The amount that Author saves in the same week is $32.
What is a simple equation?An equation which has a variable of unknown value is referred to as a simple equation. Thus, the value of the variable is to be determined.
From the given question, it can be deduced that;
Dejon saves $8 per week, while Author saves four times that of Dejon.
Thus, this implies that;
the amount that Author saves same week = 4 * $8
= $32
Therefore, the amount that Author save in the same week is $32.
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If a ball is thrown into the air with a velocity of 36 ft/s, its height (in feet) after t seconds is given by y = 36t − 16t2. Find the velocity when t = 1.
ASAPP RIGHT TOO PLS Rajindri, a physician assistant who works in an emergency room, earns $163 for every two hours that she works.
Which equation represents the relationship between d, the number of dollars Rajindri earns, and t, the amount of time Rajindri works, in hours?
A. d= 163 + t
B. d= 163/2 × t/2
C. d = 163t
D. d = 81.50t
Answer:
C
Step-by-step explanation:
The correct equation that represents the relationship between d (the number of dollars Rajindri earns) and t (the amount of time Rajindri works in hours) is: C
An admissions director wants to estimate the mean age of all students enrolled at a college. The estimate must be within 1.5 years of the population mean. Assume the population of ages is normally distributed.
(a) Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 1.8 years.
(b) The sample mean is 20 years of age. Using the minimum sample size with a 90% level of confidence, does it seem likely that the population mean could be within 10% of the sample mean? within 11% of the sample mean? Explain.
a) The minimum sample size required is 45.
b) The population mean could be within 10% of the sample mean, but less likely that it could be within 11% of the sample mean.
(a) How to determine the minimum sample size?
To determine the minimum sample size required to construct a 90% confidence interval for the population mean with a margin of error of 1.5 years, we use the formula n = (z ×σ / E)²
where n is the sample size, z is the z-score for the desired confidence level (in this case, 1.645 for a 90% confidence level), σ is the population standard deviation (1.8 years), and E is the margin of error (1.5 years).
Substituting the given values,
n = (1.645 × 1.8 / 1.5)² = 6.67² = 44.5
Rounding up to the nearest whole number, the minimum sample size required is 45.
(b) Using the minimum sample size of 45 with a 90% confidence level, the margin of error is 1.5 years. Therefore, the 90% confidence interval for the population mean is:
20 - 1.5 ≤ μ ≤ 20 + 1.5
18.5 ≤ μ ≤ 21.5
To determine whether it is likely that the population mean could be within 10% or 11% of the sample mean, we need to calculate the ranges of values that correspond to these percentages of the sample mean,
10% of 20 = 2
11% of 20 = 2.2
The range of values that are 10% of the sample mean is 20 ± 2 = 18 to 22
The range of values that are 11% of the sample mean is
20 ± 2.2 = 17.8 to 22.2
Comparing these ranges to the 90% confidence interval for the population mean, we see that:
The range of values that are 10% of the sample mean (18 to 22) is completely within the 90% confidence interval (18.5 to 21.5), so it is likely that the population mean could be within 10% of the sample mean.
The range of values that are 11% of the sample mean (17.8 to 22.2) extends slightly beyond the 90% confidence interval, so it is less likely that the population mean could be within 11% of the sample mean.
Therefore, we can conclude that it is likely that the population mean could be within 10% of the sample mean, but less likely that it could be within 11% of the sample mean.
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Determine the volume.
6 units
10 units
8 units
5
Answer:
Step-by-step explanation:
11.7
Please help me solve questions 7,8, & 9!
7) The probability of not being dealt a queen is: 12/13
8) The probability of not being dealt a 9 is: 12/13
9) The probability of not being dealt a heart is: 3/4
How to find probability of selection of cards?7) We know that in a standard deck of cards, that we have:
(4 Aces, 4 Kings, 4 Queens, 4 jacks)
Thus, probability of not selecting a queen is:
48/52 = 12/13
8) There are 4 nines in a standard deck of 52 cards. The probability of selecting the first nine is thus: 4/52.
Probability of not being dealt a 9 is: 48/52 = 12/13
9) In a standard deck of 52 cards, we know that there are 4 suits (Clubs, Hearts, Diamonds, and Spades) and there are 13 cards in each suit (Clubs/Spades are black, Hearts/Diamonds are red)
Thus, probability of not being dealt a heart = (52 - 13)/52
= 39/52
= 3/4
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11. Arrange the following number in order of magnitude starting from the smallest: ⅔, ¾, ⁵/⁷, ⅖ [A] ⅔, ¾, ⅖, ⁵/7 [B]⅖, ¾, ⅔, ⁵/7 [C]⁵/7, ⅖, ¾, ⅔ [D]⅖, ⅔, ⁵/7, ¾
The numbers arranged in the ascending order is A = 2/5 , 2/3 , 5/7 , 3/4
Given data ,
Let the numbers be represented as A
Now , the value of A is
A = { 2/3 , 3/4 , 5/7 , 2/5 }
On simplifying , we get
The value of 2/3 = 0.67
The value of 3/4 = 0.75
The value of 5/7 = 0.71
The value of 2/5 = 0.4
And , 0.4 < 0.67 < 0.71 < 0.75
Hence , the ascending order is 2/5 , 2/3 , 5/7 , 3/4
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Evaluate lim (x ^ 2 - 1)/(x ^ 3 - 1)
the limit of (x² - 1)/(x³ - 1) as x approaches 1 is equal to 2/3. got answer by try substituting x = 1 directly into the expression
what is limit of equation ?
In calculus, the limit of a function is the value that the function approaches as the input (usually denoted by x) approaches a certain value (usually denoted by a). In other words, it describes the behavior of a function as the input value gets closer and closer to a certain point.
In the given question,
To evaluate the limit of (x² - 1)/(x³ - 1) as x approaches 1, we can try substituting x = 1 directly into the expression:
(1² - 1)/(1³ - 1) = 0/0
We cannot determine the limit using direct substitution because we get an indeterminate form of 0/0.
One way to evaluate this limit is to factor the numerator and denominator and cancel out any common factors. We can factor the numerator using the difference of squares formula:
x² - 1 = (x + 1)(x - 1)
We can factor the denominator using the difference of cubes formula:
x³ - 1 = (x - 1)(x² + x + 1)
Canceling out the common factor of (x - 1) in the numerator and denominator, we get:
(x + 1)/(x² + x + 1)
Now we can substitute x = 1 directly into this expression:
(1 + 1)/(1² + 1 + 1) = 2/3
Therefore, the limit of (x² - 1)/(x³ - 1) as x approaches 1 is equal to 2/3.
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Need help finding how long it will take the object to reach 7 meters.
it should be from -4.9t^2 +29.8t+ 55 but I don't know what to do from there.
Step-by-step explanation:
7 m = - 4.9t^2 + 29.8t + 62 <====solve for 't'
0 = - 4.9t^2 + 29.8t + 56 <===== Use Quadrtatic Formula
with a = - 4.9 b = 29.8 c = 55
to find t = 7.57 seconds
ABC is an isosceles right triangle.
1). A = _____.
2). B = _____.
3). If AC = 3, then BC = _____ and AB = _____.
4). If BC = 4 then BC = _____ and AB = _____.
5). If BC = 9, then AB = ______.
6). If AB = 7 Square root 2, then BC = _____.
7). If AB = 2 square root 2, then AC = _____.
1) 45°
2) 45°
3) BC= 3 AB= sqrt(18)=3sqrt(2)
4) BC= 4 AB= sqrt(32)=4sqrt(2)
5) 9sqrt(2)
6) 7
7) 2
sqrt means square root
AC = AB because it is isosceles
pythagore theorem is used to solve 3 to 7
AB²= AC²+CB²
In 1 and 2 it is the angle in an isosceles triangle
you pick a card at random; 456. What is P (greater than 4 or less than 5) write your answer as a percentage
The probability that a card greater than 4 or less than 5 is 1 or 100%
What is probability?A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%.
The probability of picking a card greater than 4 i.e 5 or 6 = 2/3
The probability of picking a card less than 5 i.e
4 = 1/3
Therefore the probability of picking of a card greater than 4 or less than 5 = 2/3 + 1/3 = 3/3
= 1
the total probably is given by 1.
therefore in percentage , it will be 1/1 × 100 = 100%
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Every attendant at a town's Chili Cook-off received a raffle ticket. There were 9 raffle prizes,
including 7 that were gift certificates to restaurants.
If 6 prizes were randomly raffled away in the first hour of the cook-off, what is the probability
that all of them are gift certificates to restaurants?
the probability that all 6 prizes are gift certificates to restaurants is: 0.083.
What is the probability that all of them are gift certificates to restaurants?There are a total of 9 prizes, out of which 7 are gift certificates to restaurants. If 6 prizes are randomly raffled away in the first hour, there are a total of 9 choose 6 possible outcomes, or 84 possible sets of 6 prizes.
The number of outcomes in which all 6 prizes are gift certificates to restaurants is 7 choose 6, or 7 possible sets of 6 gift certificates.
Therefore, the probability that all 6 prizes are gift certificates to restaurants is:
7/84 = 0.0833 or approximately 8.33%
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Which measure do you think is more typical of the data?
The measure that is more typical of the data is the Median.
When to use median ?In comparison to the mean, utilizing the median is oftentimes a more suitable option when analyzing data sets with significant outliers or extreme values.
As these likely cause an aberrant affect on the mean, it can provide an unreliable measure of central tendency; without being subject to anomalies, the median serves as a more reliable gauge of the typical salary. Here, the Median proves significantly less than the Mean, reaffirming its superiority in cases involving considerable disparities.
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I’m stuck on this question, please help me ):
Answer:
34 m
Step-by-step explanation:
You want the width of a river as found using similar triangles.
Similar trianglesIn the attached, we have labeled the vertices of the figure and drawn it to scale. Triangles ABC and ADE are similar, so corresponding sides have the same ratio:
BE/BA = DE/DA
20.1/35 = ?/59
? = 59(20.1/35) ≈ 33.88 ≈ 34 . . . . . . multiply both sides by 59
The width of the river is about 34 meters.
__
Additional comment
The triangles are similar by the AA similarity postulate. The vertical angles are congruent, as are the right angles. You can learn to rapidly identify similar triangles and the sides that correspond. (One way to write the similarity statement is the way we did: name the vertices of the congruent angles in the same order: ∆ABC ~ ∆ADE)
The width of the river is 39m , we found by forming proportional equation because it is similar triangle.
The triangles are similar by the AA similarity postulate. The vertical angles are congruent, as are the right angles
Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion .
We have to find the width of the river
Let us form a proportional equation
x/20.1 = 59/35
Apply cross multiplication
35x=59×20.1
35x=1185.9
Divide both sides by 35
x=33.8
x=39
Hence, the width of the river is 39m , we found by forming proportional equation
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Proration means what your paying monthly
What is proration?
It's when you only pay for part of something you used.
Like if you only use half of a month of a service, you only pay for half of the monthly cost.y cost.
When is proration used?
When you sign up for a service that you don't use for a full month.
When you change your service in the middle of a billing cycle.
Why is proration used?
To make sure people only pay for what they use.
To be fair and not overcharge people for services they didn't fully use.
chatgpt
2 2/7 - 1
Must give complete and correct explanation
Answer:
1 2/7
Step-by-step explanation:
State that 1 = 7/7
2 2/7 - 7/7
Convert 2 2/7 into improper fraction
= 16/7
=16/7 - 7/7
16 - 7 = 9
= 9/7
Simplify back to mixed fraction
= 1 2/7
Help select all question asap
The following features of the circle are true:
Center: (h, k) = (- 1, 1) (Right choice: D)
Domain: - 4 ≤ x ≤ 2 (Right choice: A)
Range: - 2 ≤ y ≤ 4 (Right choice: E)
How to derive the features of the equation of a circle
Herein we find a circle represented by a equation in general form:
x² + y² + C · x + D · y + E = 0
Where A, B, C, D, E are real coefficients.
To derive the features of the circle, we need to find the standard form from the previous expression:
(x - h)² + (y - k)² = r²
Where:
(h, k) - Coordinates of the center.r - Radius of the circle.The domain and range of the equation of the circle are, respectively:
Domain: h - R ≤ x ≤ h + R
Range: k - R ≤ y ≤ k + R
First, we find the standard form by completing the square:
x² + y² + 2 · x - 2 · y - 7 = 0
(x² + 2 · x) + (y² - 2 · y) = 7
(x² + 2 · x + 1) + (y² - 2 · y + 1) = 9
(x + 1)² + (y - 1)² = 3²
Center: (h, k) = (- 1, 1)
Domain: - 4 ≤ x ≤ 2
Range: - 2 ≤ y ≤ 4
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Billy plans to invest $18,000 in a CD that compounds 1.5% monthly. He must keep his money in the CD for 10 years. How much money will he have when the investment ends?
Step-by-step explanation:
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money Billy will have when the investment ends
P = the principal amount he invested, which is $18,000
r = the annual interest rate, which is 1.5%
n = the number of times the interest is compounded per year, which is monthly or 12 times per year
t = the time period in years, which is 10 years
Plugging these values into the formula, we get:
A = 18000(1 + 0.015/12)^(12*10)
A ≈ $24,134.44
Therefore, Billy will have approximately $24,134.44 when the investment ends.
Billy can calculate his investment with the compound interest formula, using his initial investment amount, the monthly interest rate, and the number of compounding periods in 10 years. Doing so gives an end balance of approximately $20,448.24.
Explanation:Billy's investment in the CD can be calculated using the compound interest formula, which is A = P (1 + r/n)^(nt). In this formula, A represents the amount of money accumulated after n years, including interest. P is the principal amount (the initial amount of money), r is the annual interest rate (in decimal format), n is the number of times that interest is compounded per year, and t is the time in years that the money is invested for.
Plugging the given values into the formula, we have A = 18000 (1 + 0.015/12)^(12*10). Calculating this expression gives a result of approximately $20,448.24. Therefore, after 10 years, Billy will have approximately $20,448.24 in his CD.
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The Battery Park Stable feeds and houses horses used to pull tourist-filled carriages through the streets of Charleston’s historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum. The feed mixes available for the horses’ diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients (labeled as A, B, C, D, and E) needed daily to keep the average horse healthy. The table below shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes. In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he determines that 6 pounds of feed per day are the most that any horse needs to function properly.
a) (4 pts) Is this a maximization or a minimization problem? What is the objective?
b) (8 pts) Formulate this problem mathematically. Use X1, X2, and X3 to represent the amount of
oat product, enriched grain, and mineral product, respectively. Write down or type the objective
function and all constraints clearly. (Don’t forget the nonnegativity constraints when applicable.)
c) (8 pts) Set up this problem in Excel and use Solver to find the optimal daily mix of the three
feeds, i.e., the optimal value for X1, X2, and X3.
Answer:
Step-by-step explanation:
a) This is a minimization problem. The objective is to minimize the daily cost of feed for the horses while meeting their minimum nutritional requirements.
b) Let X1, X2, and X3 represent the number of pounds of oat product, enriched grain, and mineral product, respectively, that the stable owner should feed to the horses daily.
The objective function is:
Minimize 0.8X1 + 0.9X2 + 1.1X3
The constraints are:
Ingredient A: 0.15X1 + 0.20X2 + 0.25X3 >= 2.5
Ingredient B: 0.10X1 + 0.15X2 + 0.05X3 >= 2.0
Ingredient C: 0.05X1 + 0.05X2 + 0.10X3 >= 1.0
Ingredient D: 0.02X1 + 0.04X2 + 0.03X3 >= 0.3
Ingredient E: 0.08X1 + 0.07X2 + 0.06X3 >= 1.4
Non-negativity: X1 >= 0, X2 >= 0, X3 >= 0
c) To set up this problem in Excel, we can create a spreadsheet with the following columns: "Feed Mix," "Ingredient A," "Ingredient B," "Ingredient C," "Ingredient D," "Ingredient E," "Cost per Pound," and "Amount to Feed."
We can then enter the data from the table provided in the problem, and add formulas to calculate the total amount of each ingredient in each feed mix and the total cost of each feed mix based on the amount fed.
Next, we can use Solver to find the optimal mix of feed for the horses. We will set the objective to minimize the total cost of feed, subject to the constraints listed above. Solver should be able to find the optimal values of X1, X2, and X3 that meet the nutritional requirements and minimize the cost of feed.
tion K Some states now allow online gambling. As a marketing manager for a casino, you need to determine the percentage of adults in those states who gamble online. How many adults must you survey in order to be 99% confident that your estimate is in error by no more than three percentage points? Complete parts (a) and (b) below. a. Assume that nothing is known about the percentage of adults who gamble online. n= (Round up to the nearest integer.)
a) n = 4148
b) n = 2449
How to solveThe sample size for proportion:
Given that,
E = 2% = 0.02
c = 99% = 0.99
Using the z table,
= 2.576 for 99% confidene level.
Using the z-table, search for 0.9950 probability and see where is 0.9950 cumulative probability nearly and then see the corresponding z value.
Step 2/2
a)
Here p is unknown.
In this case, take p = 0.5
n = 4147.36
= 4148
b Here, p = 18% = 0.18
Now,
n = 2448.601344
= 2449
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Evaluate the function for f(2)
a. f(x) = 8x + 7
Answer:
23
Step-by-step explanation:
replace x with 2 8×2+7=23
Answer:23
Step-by-step explanation: its actually really simple, all you have to do is replace x for 2, giving you the equation: f(2) = 8x2 + 7 then all you have to do is solve, 8x2 is 16 and 16+7 is 23, and thats how you get the answer.
If m∠EFG=(3x+11)∘, and m∠GCE=(5x−23)∘, what are the measures of the central and circumscribed angles?
Responses:
m∠EFG=86∘, m∠GCE=94∘
m∠EFG=83∘, m∠GCE=97∘
m∠EFG=79∘, m∠GCE=101∘
m∠EFG=150.5∘, m∠GCE=209.5∘
The measures of the central and circumscribed angles are :
m ∠EFG=83∘, m ∠GCE=97∘
The correct option is (b)
There are two tangents on the circle C at the point E and G.
m ∠EFG=(3x+11)∘, and m ∠GCE=(5x−23)∘
Now, We have to find the measures of the central and circumscribed angles.
The line joining the center of circle to the point on circle on which there is a tangent, make an angle of 90° with the tangent itself.
∠CGF = ∠CEF = 90° ( G and F are the points on circle's tangent drawn from point F.)
Now, we can see that CGFE is a quadrilateral.
And sum of all internal angles of a quadrilateral is equal to 360°
∠C + ∠G + ∠F + ∠E = 360°
(5x - 23) + 90 + 3x + 11 + 90 = 360
=> 8x - 12 + 180 = 360
8x - 12 = 360 - 180
8x = 180 + 12
=> x = 24°
m ∠EFG = (3x + 11)°
m ∠EFG = (3× 24 + 11)° = 83°
m ∠GCE=(5x−23)∘
m ∠GCE = (5 × 24 −23)∘
m ∠GCE = 97°
The correct option is (b)
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For complete question , to see the attachment.
Change 14,277 s to h, min, and s.
Answer:
1 min = 60s
1 h = 60 mins
14277s/60s = 237 mins and 57s = 237.95 mins
237.95mins/60mins = 3 hours 57 mins 57s
= 3.97 h （cor.ti 3 sig fig.）
Answer: 237.95 minutes and 3.97 hours
Step-by-step explanation:
If you have 14,277 seconds and need to convert them into minutes, you can set up the equation 1(minute)/60(seconds) x 14,277(seconds), which equals 237. 95 minutes. To convert into hours, you just take the minutes and set up the equation 1(hour)/60(minutes) x 237.95, which equals approximately 3.966 hours. I assume your asking to convert to seconds was an accident because you started with seconds to begin with.
Use a graphing calculator to approximate the zeros and vertex of the following quadratic functions. Y = x^2 - 5x + 2
Answer:
The vertex is: [tex](\frac{5}{2} - \frac{17}{4} )[/tex]
The zero is: [tex]\frac{5+-\sqrt{17} }{2}[/tex]
Hope this helps :)
Pls brainliest...
(First, use the Pythagorean Theorem to find the value of a.)
Area = (1/2)bh OR A = bh/2
Responses
48 cm2
24 cm2
80 cm2
40 cm2
Applying the Pythagorean Theorem and the triangle area formula, the area is calculated as: b. 24 cm².
What is the Pythagorean Theorem?The Pythagorean Theorem states that the square of the longest side (hypotenuse) of a right triangle is equal to the sum of the squares of the other shorter sides or legs.
Thus, applying the Pythagorean Theorem, we have:
a = √(10² - 8²)
a = 6 cm
Base (b) = 8 cm
Height (h) = a = 6 cm
Plug in the values:
Area of triangle = 1/2 * 8 * 6
Area of triangle = 24 cm²
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ways to select the 7 math help websites
Answer:
Step-by-step explanation:
If the order you select them is important
nmber of ways = 9! / (9-7)!
= (9*8*7*6*5*4*3*2*1) / (2*1)
= 9*8*7*6*5*4*3
= 181,440.
If the order does not matter:
nmber of ways = 9! / ((9-7)! * 7!)
= 9*8 / 2*1
= 36.
Suppose that, for a certain exam, a teacher grades on a curve.
It is known that the mean is 60 and the standard deviation is 5. There are 35 students in the class. (Round your answers to the nearest whole number.)
(a) How many students should receive a C?
student(s)
(b) How many students should receive an A?
student(s
2 students to get a grade C and 0 students to get a grade A.
What is the mean?In mathematics , in addition to the mode and median, the mean is one of the measures of central tendency. the mean is the average of the values in the given set. It indicates that values in a particular data set are distributed equally.The total values provided in a datasheet must be added, and the sum must be divided by the total number of values in order to determine the mean.
What is the z-score?The relation between a value and a set of values of mean is described by the statistical measurement is known as the Z-score. Standard deviations from the mean are used to measure Z-score. A Z-score of zero means the data point of score is the same as the mean score.
We require the distribution of scores in order to estimate the number of students who should earn a given grade. We may determine the z-score for each grade using the mean and standard deviation, presuming that the scores have a normal distribution.
(a) We must locate the z-score equivalent to a C grade in order to establish the number of pupils who should receive a C. Assuming that a C equals a score between 70 and 79, the following formula can be used to determine the z-scores for these values:
(70 - 60) / 5 = 2 is the z-score for 70.
79's z-score is (79 - 60) / 5 = 3.8
We can calculate the percentage of the distribution that falls between these two z-scores using a z-table or a calculator:
P(2 ≤ z ≤ 3.8)
This implies that roughly 4.55% of the students to get a C grade. We can multiply this percentage by the total number of students Than we get
0.0455 x 35 ≈ 2
therefore,approximately 2 students to get a grade C.
b) the z-score to calculate the number of pupils who should receive an grade A. Considering that a grade A of 90 or above 90,Now we can calculate the z-score for 90
than we get .
the z-score for 90 = (90 - 60) / 5 = 6 .
We calculate the percentage of the distribution that lies to the right of this z-score using a z-table or a calculator:
P(z ≥ 6) ≈ 0
therefore,0 students to get a grade A.
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Find the value of x. Round to the nearest tenth.
The value of x is 55.6
In order to find the value of x we use sine,
sin ∅ = opposite / hypotenuse
From the question, x is the hypotenuse, the opposite is 19
So, we have
sin 20 = 19/x
x = 19/sin 20
x = 55.55
We have the answer as x = 55.6 to the nearest tenth
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Is the function g(x)=(e^x)sinb an antiderivative of the function f(x)=(e^x)sinb
We can say that it is true that the function g(x)=(e^x)sinb is an antiderivative of the function f(x)=(e^x)sinb.
How did we arrive at the solution?We can see here that to prove this, we need to show that g'(x) = f(x), where g'(x) is the derivative of g(x).
Using the product rule of differentiation, we have that:
g'(x) = (e^x)(cosb) + (sinb)(e^x).
A further simplification will give us:
g'(x) = (e^x)(cosb + sinb)
Thus, comparing g'(x) with f(x), we have: f(x) = (e^x)(sinb)
Therefore, comparing the two expressions, we can see that:
g'(x) = f(x) if and only if cosb + sinb = sinb.
This is true for all values of b, since:
cosb + sinb = sin(b + pi/4), and sin(b + pi/4) = sinb for all values of b.We can then conclude that g(x) = (e^x)sinb is indeed an antiderivative of f(x) = (e^x)sinb.
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We can say that it is true that the function g(x)=(e^x)sinb is an antiderivative of the function f(x)=(e^x)sinb.
How did we arrive at the solution?We can see here that to prove this, we need to show that g'(x) = f(x), where g'(x) is the derivative of g(x).
Using the product rule of differentiation, we have that:
g'(x) = (e^x)(cosb) + (sinb)(e^x).
A further simplification will give us:
g'(x) = (e^x)(cosb + sinb)
Thus, comparing g'(x) with f(x), we have: f(x) = (e^x)(sinb)
Therefore, comparing the two expressions, we can see that:
g'(x) = f(x) if and only if cosb + sinb = sinb.
This is true for all values of b, since:
cosb + sinb = sin(b + pi/4), and sin(b + pi/4) = sinb for all values of b.We can then conclude that g(x) = (e^x)sinb is indeed an antiderivative of f(x) = (e^x)sinb.
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In a recent year, a hospital had 4164 births. Find the mean number of births per day, then use that result and the
Poison distribution to find the probability that in a day, there are 13 births. Does it appear likely that on any given day,
there will be exactly 13 births?
Mean number of births per day ≈ 11.41.The probability of having exactly 13 births in a day,is approximately 11.41, is about 11.79%.. No, it doesnot appear likely that on any given day,there will be exactly 13 births.
Define probability?Probability can be defined as the ratio of favourable outcome to the total number of outcome.
What is Poisson distribution?A Poisson distribution is a discrete probability distribution. The chance of an event occurring a specific number of times (k) during a specific time or space period is provided by the Poisson distribution. The mean number of occurrences, denoted by the letter "lambda," is the single parameter of the Poisson distribution.
To find the mean number of births per day, we divide the total number of births in a year (4164) by the number of days in a year. Assuming a year has 365 days, the mean number of births per day would be:
Mean number of births per day = Total number of births in a year / Number of days in a year
Mean number of births per day = 4164 / 365
Mean number of births per day ≈ 11.41
Now, we can use the Poisson distribution to find the probability of having exactly 13 births in a day, given that the mean number of births per day is approximately 11.41.
The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, when the events are rare and random, and the average rate of occurrence is known.
The probability mass function (PMF) of the Poisson distribution is given by the formula:
P(X=k) = ( [tex]lambda^{k}[/tex]×e^(-λ)) / k!)
Where:
λ is the average rate of occurrence (mean) of events in the given interval
e=2.71828
k is the number of events for which we want to find the probability
k! is the factorial of k (k factorial)
In this case, the average rate of occurrence (mean) of births per day is approximately 11.41 (calculated in the previous step). So, we can plug in the values into the Poisson PMF formula:
P(X=13) = (λ¹³ ×e^(-λ)) / 13!
P(X=13) = (11.41¹³ × e^(-11.41)) / 13!
Calculating this value using a calculator or software, we can find that:
P(X=13) ≈ 0.1179 or 11.79%
So, the probability of having exactly 13 births in a day, given the mean number of births per day is approximately 11.41, is about 11.79%.
Based on this probability, it appears unlikely that on any given day there will be exactly 13 births, as the probability is relatively low. However, it is important to note that the Poisson distribution assumes that the events are rare and random, and there may be other factors that can affect the actual number of births in a day, such as seasonality, day of the week, and other external factors. Therefore, further analysis and consideration of other factors may be needed to make a more accurate assessment of the likelihood of exactly 13 births occurring in a day at a specific hospital.
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The half-life of Polonium-209 is 102 years. If we start with a sample of 108 mg of Polonium-209, determine how much will remain after 153 years.
If necessary, round answer to three decimal places.
___MG
After 153 years, 38.18 mg of Polonium-209 would remain from the initial sample of 108 mg.
How much will remain after 153 years?When we are given that half-life of Polonium-209 is 102 years, this means that after 102 years, half of the initial sample would have decayed.
We will use the below formula to calculate the amount of Polonium-209 remaining after 153 years:
= Initial amount × (1/2)^(t/half-life)
Substituting the values given in the problem, we get:
= 108 mg × (1/2)^(153/102)
= 108 mg × 0.35355339059
= 38.1837662 mg
= 38.18 mg.
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