This is the golden ratio, denoted by the Greek letter φ. It is approximately equal to 1.618.
To show that b/a-b=a/bb, we start from the equation a+b/a=a/b, which can be rearranged as follows:
[tex]a + b = a^2 / b[/tex]
Multiplying both sides by b yields:
[tex]ab + b^2 = a^2[/tex]
Subtracting ab from both sides gives:
[tex]b^2 = a^2 - ab[/tex]
Factoring out [tex]a^2[/tex] on the right-hand side gives:
[tex]b^2 = a(a - b)[/tex]
Dividing both sides by ab yields:
b/a = a/(a-b)
Substituting Φ = a/b, we have:
1/Φ = Φ/(Φ - 1)
Multiplying both sides by Φ yields:
Φ^2 - Φ - 1 = 0
This is a quadratic equation in Φ. To solve for Φ, we can use the quadratic formula:
Φ = (1 ± sqrt(5))/2
The positive root is:
Φ = (1 + sqrt(5))/2
This is the golden ratio, denoted by the Greek letter φ. It is approximately equal to 1.618.
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please help me with question 21
Using the central angle theorem, we can find the arm length of BD to be 118units.
Option C is correct.
Define central angle theorem?The angle that an arc occupies at the centre of a circle is twice as large as the angle it occupies anywhere else around the circle's circumference, according to the central angle measure theorem.
Here in the question,
Length of the arc AC = 59.
As we can see that there is a quadrilateral inscribed inside the circle.
Arc AC = 1/2 arc BD
⇒ arc BD = 2 × arc length AC
⇒ arc BD = 2 × 59
⇒ arc BD = 118.
Therefore, the length of the arc BD = 118 units.
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Using the central angle theorem, we can find the arm length of BD to be 118units.
Option C is correct.
Define central angle theorem?The angle that an arc occupies at the centre of a circle is twice as large as the angle it occupies anywhere else around the circle's circumference, according to the central angle measure theorem.
Here in the question,
Length of the arc AC = 59.
As we can see that there is a quadrilateral inscribed inside the circle.
Arc AC = 1/2 arc BD
⇒ arc BD = 2 × arc length AC
⇒ arc BD = 2 × 59
⇒ arc BD = 118.
Therefore, the length of the arc BD = 118 units.
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Find the following combinations nCr:
(a) n = 11 and r = 1.
(b) n = 11 and r = 7.
(c) n = 11 and r = 11.
(d) n = 11 and r = 4.
The following combinations nCr are:
(a) 11C1 = 11(b) 11C7 = 330(c) 11C11 = 1(d) 11C4 = 330The formula for nCr, where n is the total number of items and r is the number of items being chosen, is:
nCr = n! / (r!(n-r)!)Using this formula, we get:
(a) 11C1 = 11! / (1!(11-1)!) = 11(b) 11C7 = 11! / (7!(11-7)!) = 330(c) 11C11 = 11! / (11!(11-11)!) = 1(d) 11C4 = 11! / (4!(11-4)!) = 330So, the combinations are 11, 330, 1, and 330 for (a), (b), (c), and (d) respectively.
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In a study of hormone supplementation to enable oocyte retrieval for assisted reproduction, a team of researchers administered two hormones in different timing strategies to two randomly selected groups of women aged 36-40 years. For the Group A treatment strategy, the researchers included both hormones from day 1. The mean number of oocytes retrieved from the 98 participants in Group A was 9.7 with a 98% confidence level z-interval of (8.1, 1 1.3) Select the correct interpretation of the confidence interval with respect to the study O The researchers expect that 98% of all similarly constructed intervals will contain the true mean number of oocytes that could be retrieved from the population of women aged 36-40 years O The researchers expect that 98% of all similarly constructed intervals will contain the mean number of oocytes retrieved in the sample of 98 women aged 36-40 years O The researchers expect that the interval will contain 98% of the range of the number of oocytes retrieved in the sample of 98 women aged 36-40 years O There is a 98% chance that the the truemean number of oocytes that could be retrieved from the population of women aged 36-40 years is uniquely contained in the reported interval. O The researchers expect that 98% of all similarly constructed intervals will contain the range of the number of oocytes that could be retrieved from the population of women aged 36-40 years
The correct interpretation of the confidence interval concerning the study is that the researchers expect that 98% of all similarly constructed intervals will contain the true mean number of oocytes that could be retrieved from the population of women aged 36-40 years.
The reported interval of (8.1, 11.3) represents the range of values that is likely to contain the true mean number of oocytes retrieved from the population of women aged 36-40 years, with 98% confidence. This means that if the study were repeated multiple times with different random samples of women aged 36-40 years, and if the same statistical methods were used, then 98% of the resulting confidence intervals would contain the true population means.
It is important to note that this confidence interval applies only to the population of women aged 36-40 years, and not to other populations or age groups. Additionally, the confidence interval does not guarantee that the true population means falls within the reported interval with 98% probability, but rather that 98% of intervals constructed from repeated sampling will contain the true population means.
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11. calculate, with the assistance of eq. [10] (and showing intermediate steps), the laplace transform of the following: (a) 2.1u(t); (b) 2u(t − 1); (c) 5u(t − 2) − 2u(t); (d) 3u(t − b), where b > 0.
F (s) = ∫ e ^(-st) f(t) dt
The Laplace transforms of the given functions are:
(a) F(s) = [tex](-2.1/s) e^{(-st)} + C[/tex]
(b) F(s) = [tex]2/(s e^s)[/tex]
(c) F(s) = [tex]5 e^{(-2s)} / s - 2 / s[/tex]
(d) F(s) = [tex]3 e^{(-bs)} / s[/tex]
The Laplace transform of a function f(t) is defined as F(s) = ∫ [tex]e^{(-st)[/tex] f(t) dt, where s is a complex number. We will use this formula to find the Laplace transform of each of the given functions:
(a) 2.1u(t)
u(t) is the unit step function, which is 0 for t < 0 and 1 for t ≥ 0. Therefore, 2.1u(t) is 0 for t < 0 and 2.1 for t ≥ 0. Using the formula for the Laplace transform, we get:
F(s) = ∫ [tex]e^{(-st)[/tex] 2.1u(t) dt
= ∫ [tex]e^{(-st)[/tex] 2.1 dt (since u(t) = 1 for t ≥ 0)
= 2.1 ∫ [tex]e^{(-st)[/tex] dt
= [tex]2.1 (-1/s) e^{(-st)} + C[/tex] (using the formula ∫ [tex]e^{(-st)} dt = -1/s e^{(-st)} + C)[/tex]
= [tex](-2.1/s) e^{(-st)} + C[/tex]
(b) 2u(t − 1)
u(t − 1) is the unit step function shifted by 1 unit to the right. Therefore, u(t − 1) is 0 for t < 1 and 1 for t ≥ 1. Therefore, 2u(t − 1) is 0 for t < 1 and 2 for t ≥ 1. Using the formula for the Laplace transform, we get:
F(s) = ∫ [tex]e^{(-st)[/tex] 2u(t - 1) dt
= ∫ [tex]e^{(-s(t-1))} 2u(t - 1) d(t-1)[/tex] (using the substitution t' = t-1)
= ∫ [tex]e^{(-s(t-1))} 2 d(t-1)[/tex] (since u(t - 1) = 1 for t ≥ 1)
= 2 ∫ [tex]e^{(-s(t-1))} d(t-1)[/tex]
= [tex]2 e^{(-s(t-1))} / -s[/tex] | from 1 to infinity
= [tex]2/(s e^s)[/tex]
(c) 5u(t − 2) − 2u(t)
Using linearity, we can find the Laplace transform of each term separately and then subtract them:
F(s) = L{5u(t − 2)} - L{2u(t)}
= 5 L{u(t − 2)} - 2 L{u(t)}
= [tex]5 e^{(-2s)} / s - 2 / s[/tex]
(d) 3u(t − b), where b > 0
Using a similar approach as in (b) and (c), we get:
F(s) = 3 L{u(t − b)}
= [tex]3 e^{(-bs)} / s[/tex]
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The like terms in the box are: -2x and 21x 21x and -14 3x2 and -2x
Based on the list of options, the like terms in the box are: -2x and 21x
Identifying the like termsAn expression can be simplified by combining like terms.
Like terms are those that have the same variable and exponent, so they can be combined by adding or subtracting their coefficients.
In the list of options, there are terms that have the variable x:
Of these, the terms 21x and -2x are like terms because they have the same variable x, but with different coefficients. Therefore, we can combine them by adding their coefficients:
21x - 2x = 19x
Similarly, there are two terms that do not have the variable x: 3x^2 and -14.
These are not like terms because they do not have the same variable or exponent.
Therefore, we cannot combine them further.
Therefore, the like terms in the given expression are -2x and 21x, and they can be combined to get 19x.
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Find the area of the shape below.
3.) The area of the shape would be = 77mm².
How to determine the area of the given shape ?To determine the area of the given shape, the area of the trapezium and the rectangule is both calculated and summed up
The area of a rectangle = length×width
width = 5 mm
length = 10mm
area = 10×5 = 50 mm²
Area of trapezium = ½(a+b)h
where;
a = 10mm
b = 8mm
h = 8-5 = 3mm
area = 1/2(10+8)×3
= 54/2 = 27mm²
Therefore area of the shape = 50+27 = 77mm².
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1. construct a 95onfidence interval to estimate the population mean using the following data: sample mean = 75 population standard deviation = 20 sample size = 36
95% confidence interval for the population mean is (67.35, 82.65).
How to construct a 95% confidence interval for the population mean?We can use the formula:
CI = x⁻ ± z*(σ/√n)
where x⁻ is the sample mean, σ is the population standard deviation, n is the sample size, and z is the critical value from the standard normal distribution corresponding to the desired confidence level (95% in this case).
First, we need to find the value of z. The area under the standard normal distribution curve between -z and z is 0.95. Using a table or calculator, we find that the critical value for a 95% confidence level is 1.96.
Now we can plug in the values we have:
CI = 75 ± 1.96*(20/√36)
= 75 ± 7.65
Therefore, the 95% confidence interval for the population mean is (67.35, 82.65). We can be 95% confident that the true population mean lies within this interval
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A Bloomberg BusinessWeek subscriber study asked, "In the past 12 months, when traveling for business, what type of airline ticket did you purchase most often?" A second question asked if the type of airline ticket purchased most often was for domestic or international travel. Sample data obtained are shown in the following table.
a. Using a .05 level of significance, is the type of ticket purchased independent of the type of flight?
What is your conclusion?
b. Discuss any dependence that exists between the type of ticket and type of flight.
Null hypothesis: The type of airline ticket purchased is independent of the type of flight.
Alternative hypothesis: The type of airline ticket purchased is dependent on the type of flight.
how to determine type of flight?To determine whether the type of airline ticket purchased is independent of the type of flight, we can use the chi-square test for independence. The invalid speculation is that the two factors are free, while the elective speculation is that they are reliant.
a. Speculations:
Negative hypothesis: The sort of aircraft ticket bought is free of the kind of flight.
Other possibilities: The kind of flight determines which kind of airline ticket is purchased.
To summarize the data, we can make a contingency table as follows:
Domestic International Total
Business class 85 78 163
Economy class 157 130 287
First class 20 22 42
Total 262 230 492
We calculate a p-value of 0.150 and a test statistic of 3.794 using the chi-square test for independence. The critical value for a chi-square distribution with two degrees of freedom is 5.991 at a significance level of 0.05.
We are unable to reject the null hypothesis because the calculated test statistic (3.794) is lower than the critical value (5.991). We don't have enough evidence to say that the kind of airline ticket you buy depends on the type of flight.
b. According to the contingency table, the majority of tickets purchased were for domestic and international flights in economy class. However, compared to domestic flights, business class tickets are purchased slightly more frequently on international flights. On the other hand, compared to international flights, tickets in economy class are purchased slightly more frequently for domestic flights. These distinctions are not genuinely critical, however they really do recommend a few reliance between the kind of ticket and the sort of flight. This could be because of things like how long the flight is or what the trip is for.
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triangle def is circumscribed about circle o with de=15 df=12 and ef=13
Find the length of each segment whose endpoints are D and the points of tangency on DE and DF
Answer:
7
Step-by-step explanation:
You want the tangent lengths from point D for ∆DEF circumscribing a circle, given DE=15, DF=12, DF=13.
Tangent segmentsThe lengths of the tangent segments from vertex D are ...
d = (DE +DF -EF)/2 = (15 +12 -13)/2 = 7
The tangent segments with end point D are 7 units long.
__
Additional comment
The tangents from each point are the same length, so we have ...
d + e = DE . . . . where d, e, f are the lengths of the tangents from D, E, F
e + f = EF
d + f = DF
Forming the sum shown above, we have ...
DE +DF -EF = (d +e) +(d +f) -(e +f) = 2d
d = (DE +DF -EF)/2 . . . . as above
The other tangents are e = 8, f = 5.
find the indefinite integral. (use c for the constant of integration.) 4t 1 − 16t4 dt
The indefinite integral of 4t(1-16t^4) dt is: 2t^2 - (4/5)t^6 + c, Here, C is the constant of integration, which can be written as C = C1 + C2.
To find the indefinite integral of the given function, we'll integrate term by term. The given function is:
∫(4t - 16t^4) dt
Now we'll integrate each term:
∫4t dt - ∫16t^4 dt
For the first term, the power rule for integration states that ∫t^n dt = (t^(n+1))/(n+1) + C, where n is a constant:
∫4t dt = 4∫t^1 dt = 4(t^(1+1))/(1+1) + C1 = 4t^2/2 + C1 = 2t^2 + C1
For the second term, we'll apply the same rule:
∫16t^4 dt = 16∫t^4 dt = 16(t^(4+1))/(4+1) + C2 = 16t^5/5 + C2 = (16/5)t^5 + C2
Now combine the results:
∫(4t - 16t^4) dt = 2t^2 + (16/5)t^5 + C
Here, C is the constant of integration, which can be written as C = C1 + C2.
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Charlie bought shares worth £7000.
a) After one month, their value had increased by 12%. How much were
they worth after one month?
b) After two months, this new value had decreased by 15%. How much
were they worth after two months?
Give your answers in pounds
Answer:
a) After one month, the value of the shares increased by:
£7000 x 12/100 = £840
Therefore, the shares were worth:
£7000 + £840 = £7840
b) After two months, the value of the shares decreased by:
£7840 x 15/100 = £1176
Therefore, the shares were worth:
£7840 - £1176 = £6664
Help me pls it’s extra credit I need it
Answer: A
Step-by-step explanation:
y int = b= 4
slope=rise/run= rise of 4/ run of 2 = 2
so
y=2x+4
Researchers found from of a random sample of n=1522 adults in the US who were asked whether they consider a gym membership to be a necessity or a luxury that the proportion of those who answered "necessity" is 0.15 with a margin of error of 0.02 What is the correct calculation for a 95% confidence interval for the true proportion of all US adults who feel a gym membership is a necessity?
A. 0.15 - 2x 0.02 to 0.15 + 2 x 1522 0.02 71522
B. 0.15 - 2 x 0.02 to 0.15 + 2 x 0.02
C. 0.15 -0.02 to 0.15 + 0.02
D. 0.15 - 0.02 1522 to 0.15 + 0.02 V1522
The correct answer is option C i.e. 0.15 - 0.02 to 0.15 + 0.02
How to calculate 95% confidence interval?The correct calculation for a 95% confidence interval for the true proportion of all US adults who feel a gym membership is a necessity is:
Margin of error = z√(p(1-p)/n)
where z is the z-score corresponding to the desired level of confidence (95% in this case), p is the sample proportion (0.15), and n is the sample size (1522).
From a standard normal distribution table, the z-score for a 95% confidence level is approximately 1.96.
Substituting these values into the formula, we get:
Margin of error = 1.96 * √(0.15*(1-0.15)/1522) ≈ 0.02
Therefore, the 95% confidence interval is:
0.15 - 0.02 to 0.15 + 0.02
which simplifies to: [0.13, 0.17]
So, the correct answer is option C.
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The area of the shaded region is 20cm².
Find the value of x, correct to 3 significant figures.
The value of x is 8.37
What is the area of the shaded region?
The area of the shaded area is the difference between the total area of the polygon and the area of the portion of the polygon that is not shaded. In polygons, the area of the shaded component might appear in two different ways. A polygon's sides or its center are both potential locations for the shaded area.
Here, we have
Given: The area of the shaded region is 20cm².
we have to find the value of x.
x in this case is the radius. In fact, both the height and base are the radius.
To find the radius, we need to form an equation. The only info given is with the area of the shaded area which is 20cm².
The area of the sector - an area of the triangle = the shaded area.
Area of the sector = πr²/4
Area of triangle = (1/2)bh
Area of the triangle = x²/2
The area of the triangle is in that way, as the height and base are x ( and x is the radius here!)
= πr²/4 - x²/2
Multiply 4 with the whole equation as it is the LCM.
= 4(πr²/4 - x²/2) = 80
= πx² - 2x² = 80
= 1.142x² = 80
x² = 70.1
x = 8.37
Hence, the value of x is 8.37
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find the area enclosed by the ellipse x 2 a 2 y 2 b 2 = 1 us
The value of the area is πab which is enclosed by the ellipse with the equation (x²/a²) + (y²/b²) = 1.
To find the area enclosed by the ellipse with the equation (x²/a²) + (y²/b²) = 1.
To find the area of this ellipse, use the formula A = πab, where A is the area, a is the semi-major axis, and b is the semi-minor axis.
First, identify the values of a and b from the given equation.
In the equation (x²/a²) + (y²/b²) = 1, a² is the coefficient of x², and b² is the coefficient of y².
Now, calculate the area using the formula A = πab.
Plug the values of a and b into the formula and multiply them with π to find the area.
So, the area enclosed by the ellipse (x²/a²) + (y²/b²) = 1 is A = πab.
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Question 4(Multiple Choice Worth 2 points)
(Appropriate Measures MC)
A charity needs to report its typical donations received. The following is a list of the donations from one week. A histogram is provided to display the data.
10, 11, 35, 39, 40, 42, 42, 45, 49, 49, 51, 51, 52, 53, 53, 54, 56, 59
A graph titled Donations to Charity in Dollars. The x-axis is labeled 10 to 19, 20 to 29, 30 to 39, 40 to 49, and 50 to 59. The y-axis is labeled Frequency. There is a shaded bar up to 2 above 10 to 19, up to 2 above 30 to 39, up to 6 above 40 to 49, and up to 8 above 50 to 59. There is no shaded bar above 20 to 29.
Which measure of variability should the charity use to accurately represent the data? Explain your answer.
The range of 13 is the most accurate to use, since the data is skewed.
The IQR of 49 is the most accurate to use to show that they need more money.
The range of 49 is the most accurate to use to show that they have plenty of money.
The IQR of 13 is the most accurate to use, since the data is skewed.
Answer:
The IQR of 13 is the most accurate to use, since the data is skewed. The reason for this is that the data is not evenly distributed, as shown by the histogram with a large number of donations in the higher range. The IQR is a measure of variability that is less sensitive to outliers and skewed data than the range, which makes it a better choice for this type of data. Additionally, the IQR can provide information on the spread of the middle 50% of the data, which can be useful in understanding the typical donation range for the charity.
compute the area bounded by the circle =2 and the rays =5, and = as an integral in polar coordinates. (use symbolic notation and fractions where needed.)
The area bounded by the circle =2 and the rays =5, and = is 4π/3 square units.
To compute the area bounded by the circle =2 and the rays =5, and = as an integral in polar coordinates, we can use the formula:
A = (1/2)∫[b,a] r² dθ
where r is the polar radius, and a and b are the angles where the rays intersect the circle.
Since the circle has a radius of 2, we have r = 2 for the equation of the circle. We also know that the rays intersect the circle at angles π/3 and 5π/3 (or 2π/3 and 4π/3 in the standard position).
Therefore, we have:
A = (1/2)∫[2π/3,4π/3] (2)² dθ
A = 2∫[2π/3,4π/3] dθ
A = 2(4π/3 - 2π/3)
A = 2(2π/3)
A = 4π/3
So, the area bounded by the circle =2 and the rays =5, and = is 4π/3 square units.
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if f ◦ g is onto, must g be onto? explain your answer
f ◦ g is onto, it does not guarantee that g must be onto.
If f ◦ g is onto, must g be onto? The answer is no, g does not necessarily have to be on. Let's explain this with the following steps:
1. Definition of ontological (surjective) function: A function g: A → B is onto if for every element b in the codomain B, there exists at least one element a in the domain A such that g(a) = b.
2. Definition of function composition (fg): Given two functions f: B → C and g: A → B, the composition f ◦ g: A → C is a function such that (f ◦ g)(a) = f(g(a)) for all an in A.
3. Given that f ◦ g is on, for every element c in the codomain C, there exists at least one element a in the domain A such that (f ◦ g)(a) = c.
4. However, the surjectivity of f ◦ g does not imply the surjectivity of g. This is because f may "compensate" for any lack of subjectivity in g. In other words, even if g does not map to every element in its codomain B, f might still map the outputs of g to every element in its codomain C.
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Find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p. n=121, p=0.62 The mean, h, is (Round to the nearest tenth as needed.)
The mean is approximately 75.0, the variance is approximately 28.9, and the standard deviation is approximately 5.4.
How to find the mean, variance, and standard deviation?To find the mean, variance, and standard deviation of a binomial distribution with n = 121 and p = 0.62, you can use the following formulas:
1. Mean (μ) = n * p
2. Variance (σ²) = n * p * (1 - p)
3. Standard Deviation (σ) = √(variance)
Step 1: Calculate the mean.
Mean (μ) = n * p = 121 * 0.62 ≈ 75.02
Step 2: Calculate the variance.
Variance (σ²) = n * p * (1 - p) = 121 * 0.62 * (1 - 0.62) ≈ 28.91
Step 3: Calculate the standard deviation.
Standard Deviation (σ) = √(variance) = √(28.91) ≈ 5.38
So, the mean is approximately 75.0, the variance is approximately 28.9, and the standard deviation is approximately 5.4.
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The price of entrées at fast food restaurants in the area have an unknown distribution with a mean price of $6.75 and a standard deviation of $1.08. If you randomly select 45 combo meals around town, what is the probability that their average price will be less than $6.50?
The probability that the average price of 45 randomly selected combo meals around town will be less than $6.50 can be calculated using the central limit theorem.
According to the central limit theorem, the sampling distribution of the sample mean becomes approximately normal, regardless of the distribution of the population, if the sample size is large enough (n > 30).
Therefore, we can assume that the sample mean of the 45 combo meals follows a normal distribution with a mean of $6.75 and a standard deviation of $1.08/sqrt(45) = $0.161.
To find the probability that the sample mean is less than $6.50, we need to standardize the distribution using the z-score formula:
z = (x - μ) / (σ / sqrt(n))
where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Plugging in the values, we get:
z = (6.50 - 6.75) / (1.08 / sqrt(45)) = -1.73
Looking up the z-score in the standard normal distribution table, we find that the probability of a z-score less than -1.73 is approximately 0.04.
Therefore, the probability that the average price of 45 randomly selected combo meals around town will be less than $6.50 is 0.04 or 4%.
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Find the absolute extrema of the function on the closed interval.
y= 2-|t-2|, [-9,3]
minimum (x,y) = __
maximum (x,y) = ___
The absolute minimum is (-9, -9) and the absolute maximum is (2, 2).
How to find the absolute extrema of the function?To find the absolute extrema of the function y = 2 - |t - 2| on the closed interval [-9, 3], we need to follow these steps:
1. Identify the critical points: These are the points where the derivative is zero or undefined.
2. Evaluate the function at the critical points and endpoints of the interval.
3. Compare the function values to find the minimum and maximum.
Step 1: Critical points
The derivative of the absolute value function is undefined at t = 2. Thus, the critical point is t = 2.
Step 2: Evaluate the function at the critical point and endpoints
Evaluate the function at t = -9, 2, and 3.
At t = -9, y = 2 - |-9 - 2| = 2 - 11 = -9.
At t = 2, y = 2 - |2 - 2| = 2 - 0 = 2.
At t = 3, y = 2 - |3 - 2| = 2 - 1 = 1.
Step 3: Compare the function values
The minimum value is -9 at t = -9 and the maximum value is 2 at t = 2.
Therefore, the absolute minimum is (-9, -9) and the absolute maximum is (2, 2).
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find the absolute maximum and absolute minimum values of f on the given interval. f(t) = t − 3√ t , [−1, 4]
The absolute maximum value of f(t) is approximately -0.1213 at t = 9/4, and the absolute minimum value is -2 at t = 4.
To find the absolute maximum and absolute minimum values of f on the given interval [−1, 4], we first need to find the critical points of the function f(t) = t − 3√t.
Taking the derivative of f(t) with respect to t, we get:
f'(t) = 1 - (3/2)t^(-1/2)
Setting f'(t) = 0 to find critical points, we get:
0 = 1 - (3/2)t^(-1/2)
(3/2)t^(-1/2) = 1
t^(-1/2) = 2/3
t = (2/3)^(-2) = 2.25
So the only critical point of f(t) on the interval [−1, 4] is t = 2.25.
Now we need to evaluate f(t) at the endpoints of the interval and at the critical point to determine the absolute maximum and minimum values of f on the interval:
f(-1) = -1 - 3√(-1) = -1 - 3i
f(4) = 4 - 3√4 = 4 - 6 = -2
f(2.25) = 2.25 - 3√2.25 = 2.25 - 3(1.5) = -2.25
Therefore, the absolute maximum value of f on the interval [−1, 4] is f(-1) = -1 - 3i, and the absolute minimum value of f on the interval is f(4) = -2.
To find the absolute maximum and minimum values of f(t) = t - 3√t on the interval [-1, 4], we need to evaluate the function at its critical points and endpoints.
First, we find the critical points by taking the derivative of the function and setting it to zero:
f'(t) = 1 - (3/2)t^(-1/2)
To solve for critical points, set f'(t) = 0:
0 = 1 - (3/2)t^(-1/2)
(3/2)t^(-1/2) = 1
t^(-1/2) = 2/3
t = (2/3)^(-2) = 9/4
Since 9/4 is within the interval [-1, 4], it is a valid critical point.
Now, evaluate the function at the critical point and the endpoints:
f(-1) = -1 - 3√(-1)
(Note: This value is complex, and we're looking for absolute max/min in the real domain, so we'll ignore this endpoint)
f(9/4) = (9/4) - 3√(9/4) ≈ -0.1213
f(4) = 4 - 3√4 = -2
So, the absolute maximum value of f(t) is approximately -0.1213 at t = 9/4, and the absolute minimum value is -2 at t = 4.
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Given matrices A and B shown below, find 3B - 6A.
The result of the expression (3B - 6A) for given matrices A and B will be:[tex]\begin{pmatrix}-36 \\-3 \\3 \\\end{pmatrix}[/tex]
What is 'Matrix' in mathematics?A rectangular array of numbers, symbols, or expressions arranged in rows and columns is known in mathematics as a matrix..It is normally marked by a capital letter, and the matrix elements are usually wrapped in brackets or brackets. Matrices are useful in many mathematical subjects, including linear algebra, calculus, statistics, and computer science.
A matrix with "m" rows and "n" columns is said to have "mxn" dimensions (pronounced as "m by n").
For the given problem,
[tex]\[3B = \begin{pmatrix}3(-6) \\3(1) \\3(1) \\\end{pmatrix}= \begin{pmatrix}-18 \\3 \\3 \\\end{pmatrix}\][/tex]
[tex]\[6A = \begin{pmatrix}6(3) \\6(1) \\6(0) \\\end{pmatrix}= \begin{pmatrix}18 \\6 \\0 \\\end{pmatrix}\][/tex]
[tex]\[3B - 6A = \begin{pmatrix}-18 \\3 \\3 \\\end{pmatrix}- \begin{pmatrix}18 \\6 \\0 \\\end{pmatrix}= \begin{pmatrix}-18 - 18 \\3 - 6 \\3 - 0 \\\end{pmatrix}= \begin{pmatrix}-36 \\-3 \\3 \\\end{pmatrix}\][/tex]
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Find the exact location of all the relative and absolute extrema of the function. (Order your answers from smallest to largest x.)g(x) = 9x2 − 36xg has a relative maximum at (x,y):g has an absolute minimum at (x,y):
g has a relative minimum and absolute minimum at (x,y) = (2, -36).
To find the exact location of all the relative and absolute extrema of the function g(x) = 9x^2 - 36x, follow these steps,
1. Find the derivative of g(x) with respect to x:
g'(x) = d(9x^2 - 36x)/dx = 18x - 36
2. Set g'(x) equal to 0 to find critical points:
18x - 36 = 0
18x = 36
x = 2
3. Determine the type of extrema (minimum or maximum) by analyzing the second derivative:
g''(x) = d^2(9x^2 - 36x)/dx^2 = 18
Since g''(x) is positive at x = 2, there is a relative minimum at this point.
4. Calculate the value of g(x) at the critical point:
g(2) = 9(2)^2 - 36(2) = 9(4) - 72 = 36 - 72 = -36
5. Summarize the findings:
g has a relative minimum at (x,y) = (2, -36), and since this is the only extrema of the function, it is also an absolute minimum.
g has a relative minimum and absolute minimum at (x,y) = (2, -36).
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[7/2+(4/2)]+3/5 verify the associative property of addition for the following rational numbers
Left-hand side = 61/10.
Right-hand side = 51/10.
The left-hand side is not equal to the right-hand side, we can see that the associative property of addition does not hold for the given rational numbers.
What are rational exponents?
Rational exponents are exponents that are expressed as fractions.
To verify the associative property of addition for the given rational numbers, we need to check if:
(7/2 + (4/2)) + (3/5) = 7/2 + ((4/2) + (3/5))
First, let's simplify each side of the equation:
Left-hand side:
(7/2 + (4/2)) + (3/5)
= (11/2) + (3/5)
= (55/10) + (6/10)
= 61/10.
Right-hand side:
7/2 + ((4/2) + (3/5))
= 7/2 + (8/5)
= (35/10) + (16/10)
= 51/10.
Since the left-hand side is not equal to the right-hand side, we can see that the associative property of addition does not hold for the given rational numbers.
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A building is constructed using bricks that can be modeled as right rectangular prisms with a dimension of 8/ in by 3 in by 3 in. If the bricks cost $0.07 per cubic inch, find the cost of 300 bricks. Round your answer to the nearest cent.
The cost of 300 bricks is equal to $1,512.
How to calculate the volume of a rectangular prism?In Mathematics and Geometry, the volume of a rectangular prism can be calculated by using the following formula:
Volume of a rectangular prism = L × W × H
Where:
L represents the length of a rectangular prism.W represents the width of a rectangular prism.H represents the height of a rectangular prism.By substituting the given dimensions (parameters) into the formula for the volume of a rectangular prism, we have;
Volume of bricks = 8 × 3 × 3
Volume of bricks = 72 cubic inches.
For the cost per cubic inch, we have:
Cost per cubic inch = 72 × 0.07
Cost per cubic inch = $5.04
For the cost of 300 bricks, we have:
Cost of 300 bricks = 300 × $5.04
Cost of 300 bricks = $1,512
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The demand function for a company's product is p = 26e^−0.6q where q is measured in thousands of units and p is measured in dollars.
(a) What price should the company charge for each unit in order to sell 6500 units? (Round your answer to two decimal places.)
$__________
(b) If the company prices the products at $6.50 each, how many units will sell? (Round your answer to the nearest integer.)
__________units
(a) To find the price the company should charge for each unit to sell 6,500 units, we need to substitute q with 6.5 (since q is measured in thousands of units) in the demand function p = 26e^(-0.6q): p = 26e^(-0.6 * 6.5)
After calculating, we get: p ≈ $2.98
So, the company should charge approximately $2.98 per unit to sell 6,500 units.
(b) To find how many units will sell if the company prices the products at $6.50 each, we need to solve for q in the demand function p = 26e^(-0.6q) with p = $6.50: 6.50 = 26e^(-0.6q)
Now, we need to solve for q: q = ln(6.50/26) / -0.6 ≈ 1.884
Since q is measured in thousands of units, the company will sell approximately 1,884 units when the price is $6.50 each.
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A 9th order, linear, homogeneous, constant coefficient differential equation has a characteristic equation which factors as follows.
(r^2+6r+10)^2r^2(r-1)^3=0
Write the nine fundamental solutions to the differential equation. Use t as the independent variable.
The nine fundamental solutions to the differential equation are:
[tex]e^{(-3+i)t}, e^{(-3-i)t}, e^t, te^t,[/tex] 1, t, t²/2!, t³/3!, [tex]t^4[/tex]/4!, [tex]e^{(-5+i)t}, ~and ~e^{(-5-i)t}[/tex]
We have,
The characteristic equation of the given differential equation is:
[tex](r^2 + 6r + 10)^2 \times r^2 (r - 1)^3 = 0[/tex]
We can find the fundamental solutions by looking at the roots of the characteristic equation.
The roots can be categorized as follows:
Roots of multiplicity 2 = -3 + i and -3 - i
Roots of multiplicity 2 = 1
Root of multiplicity 1 = 0
Root of multiplicity 2 = -5 + i and -5 - i
For each of these roots, we need to find the corresponding fundamental solution.
For the roots (-3 + i) and (-3 - i), the corresponding fundamental solutions are:
[tex]e^{(-3+i)t}~ and~ e^{(-3-i)t}[/tex]
For root 1, the corresponding fundamental solutions are:
[tex]e^t~and~te^t[/tex]
For the root 0, the corresponding fundamental solutions are:
1, t, t²/2!, t³/3!, ..., [tex]t^8[/tex]/8!
For the roots (-5 + i) and (-5 - i), the corresponding fundamental solutions are:
[tex]e^{(-5+i)t} ~and~e^{(-5-i)t}[/tex]
Therefore,
The nine fundamental solutions to the differential equation are:
[tex]e^{(-3+i)t}, e^{(-3-i)t}, e^t, te^t,[/tex] 1, t, t²/2!, t³/3!, [tex]t^4[/tex]/4!, [tex]e^{(-5+i)t}, ~and ~e^{(-5-i)t}[/tex]
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a student spends 14 hour on a project on wednesday and 18 hour on the same project on thursday. the student tells the teacher the project took 12 hour to complete.which statement is true?responses the student is correct because 78 is greater than 12.the student is correct because 7 8 is greater than 1 2 .the student is correct because 78 is less than 12.the student is correct because 7 8 is less than 1 2 .the student is incorrect because 38 is greater than 12.the student is incorrect because 3 8 is greater than 1 2 .the student is incorrect because 38 is less than 12.
The correct statement is: the student is incorrect because 32 is less than 12.
This is because the student spent a total of 32 hours on the project (14 on Wednesday + 18 on Thursday), but claimed it took only 12 hours to complete. Therefore, the student's statement is not true.
The word "more" is used when one number is greater than another. Use more even when comparing two weights. For example, Joe went to the ice cream parlor. She likes the chocolate and vanilla flavor of the ice cream but wants to buy a cheaper ice cream cone. He asked the price of the chocolate and vanilla cones.
The seller shows the price of two types of cones: $10 for a dough cone and $5 for a vanilla cone. Then to compare the value of the two cones, Joe should use the concept of "more". As we can see, the chocolate cone is more expensive than the vanilla cone. The price of the cookie ($10) is more than the price of the vanilla cone ($5), so 10 > 5. So he compares the price of two ice creams and decides to buy the vanilla ice cream.
Just like two weights, distance, volume etc. Use the greater than sign as compare.
The student is incorrect because 38 is greater than 12. This is because the student spent 14 hours on Wednesday and 18 hours on Thursday, which totals 14 + 18 = 38 hours, and 38 hours is greater than the reported 12 hours.
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A rectangular prism has a height of 22 yards and a base with an area of 152 square yards. What is its volume?
Answer:
3344 cubic yards
Step-by-step explanation:
The volume of a rectangular prism is length x width x height.
If the area of the base is 152, that means the length x width = 152
So, 152 x 22 = 3344.