The vertices of the triangle after reflection is
D)N′(−5, 2), M′(−8, 1), O′(−7, 3).
What is triangle?
A triangle is a form of polygon with three sides; the intersection of the two longest sides is known as the triangle's vertex. There is an angle created between two sides. One of the crucial elements of geometry is this
Remember that the general rule to reflect over a vertical line in the form
if x = a then,
=> (x , y) -> (-x-2a , y)
For x = 5, we'll have that the general rule is:
=> (x , y) -> (-x-10 , y).
Now the triangle vertices are,
N(-5,2) => (-(-5)-10,2) => (5-10 , 2) => N'(-5,2)
M(-2,1) => (-(-2)-10,1) => (2-10,1) => M' (-8,1)
O(-3,3) => (-(-3)-10,3) => (3-10,3) => O'(-7,3)
Hence the vertices of the triangle after reflection is D)N′(−5, 2), M′(−8, 1), O′(−7, 3).
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8.G.C.9
Which formula will find my volume?
The formula to find the volume of attached figure: V = πr²h
The correct answer is an option (d)
In the attached image we can observe that the figure is of cylinder with radius 'r' and height 'h'
This means that we need to find the volume of cylinder.
We know that the formula for tha cylinder is:
Volume of cylinder = Base area × height of cylinder
As we know that the base of cylinder is circular in shape.
so, the base area of cylinder would be,
A = πr²
when we say height of the cylinder then it means the perpendicular distance between two parallel bases of cylinder. It is also known as length of the cylinder.
So, the formula for volume would be,
V = πr²h
Therefore, the correct answer is an option (d)
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use your own words to explain what a sequence is. Give an example to illustrate this explanation 
A sequence is a collection of numbers in which a pattern between consecutive numbers is established.
One example is a geometric sequence with first term of 2 and common ratio of 3, which is:
2, 6, 18, 54, ...
What is a geometric sequence?A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio q.
Considering a geometric sequence with first term of 2 and common ratio of 3, the other terms are given as follows:
2 x 3 = 6.6 x 3 = 18.18 x 3 = 54.And the pattern continues for however many terms are there in the sequence.
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Determine whether the series is absolutely convergent, conditionally convergent, or divergent.[infinity] (−1)nnn3 + 5n = 1(-1)^n (n/sqrt n^3+5)absolutely convergentconditionally convergentdivergent
The given series is conditionally convergent.
We can use the alternating series test to show that the series converges. First, we can rewrite the terms of the series as:
an = (-1)ⁿ * (n/√(n³ + 5))The terms of the series are decreasing in absolute value and approach zero as n approaches infinity. Also, the series is alternating in sign, so we can apply the alternating series test. Therefore, the series converges.
To determine whether the series is absolutely convergent or conditionally convergent, we need to check the convergence of the series of absolute values:
∑ |an| = ∑ (n/√(n³ + 5))We can use the limit comparison test to compare this series with the series ∑ (1/√(n)). We have:
lim (n/√(n³ + 5)) / (1/√(n)) = lim (n*√(n)) / √(n³ + 5) = lim 1 / √(1 + 5/n²) = 1
Since this limit is a positive finite number, the series ∑ |an| and the series ∑ (1/√(n)) have the same behavior. The series ∑ (1/√(n)) is a p-series with p=1/2, which is known to be divergent. Therefore, the series ∑ |an| is also divergent. Since the original series is convergent but |an| is divergent, the original series is conditionally convergent.
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Find the solution of the differential equation that satisfies the given initial condition. dy 5xe, y(0) = 0 dx -In + = -5x² 2 X
The solution of the given differential equation that satisfies the initial condition y(0) = 0 is 0 = (5/2)(0)² - (1/4)(ln(0))^2 - (5/3)(0)³ + C.
To find the solution of the given differential equation that satisfies the initial condition y(0) = 0, we will follow these steps,
1. Identify the differential equation: dy/dx = 5x - (ln(x)/2) - 5x²
2. Integrate both sides of the equation with respect to x.
Integral of dy = Integral of (5x - (ln(x)/2) - 5x²) dx
Since y(0) = 0, we have:
y(x) = Integral of (5x - (ln(x)/2) - 5x²) dx
3. Perform the integration:
y(x) = (5/2)x² - (1/4)(ln(x))^2 - (5/3)x³ + C
4. Determine the value of the constant C using the initial condition y(0) = 0:
0 = (5/2)(0)² - (1/4)(ln(0))^2 - (5/3)(0)³ + C
Since ln(0) is undefined, we cannot solve for C using the initial condition y(0) = 0. However, the given initial condition is not consistent with the differential equation, so there may be an error in the problem statement.
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A squre is cut into three rectangles X, Y and Z
The algebraic expression for the width of rectangle X is:
w = (s - n - 5)/2
Since a square has four equal sides, each side of the square can be represented as s. The area of the square is s². When it is cut into three rectangles X, Y and Z, the area of the square is equal to the sum of the areas of the three rectangles.
So, we have:
s² = 10w + 5n + 5(s-n-5)/2
Simplifying this equation, we get:
s² = 10w + 5n + (5s - 5n - 25)/2
Multiplying both sides by 2, we get:
2s² = 20w + 10n + 5s - 5n - 25
Simplifying this equation, we get:
20w = 2s² - 10n - 5s + 5n + 25
Dividing both sides by 2 and rearranging the terms, we get the algebraic expression for the width of rectangle X:
w = (s - n - 5)/2
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Complete Question:
A square is cut into three rectangles X, Y and Z. Rectangle X has length 10cm. Rectangle Y has length n cm and width 5cm. Write down an algebraic expression for the width of rectangle X.
a population has a mean μ=80 and a standard deviation σ=7. find the mean and standard deviation of a sampling distribution of sample means with sample size n=49.
The population mean is equal to 80 for the mean of a sampling distribution of sample means with n=49, and the standard deviation is equal to 1/√(n).
What does standard deviation mean?The standard deviation is a statistician's gauge of a group of values' degree of dispersion or variation. While a high standard deviation suggests that the values are dispersed over a wider range, a low standard deviation suggests that the values tend to be close to the mean (also known as the expected value) of the set.With a sample size of 49, the mean of a sampling distribution of sample means is equal to the population mean of 801.
With a sample size of n=49, the standard deviation of a sampling distribution of sample means is identical.
n=49 is equal to σ/√(n)
= 7/√(49) = 1¹
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work out the value of (3squared)squared times (10cubed)squared
Answer:
Scientific Notation:
[tex]8.1*10^7[/tex]
Expanded form:
81000000
Hope this helps :)
Pls brainliest...
The value of ''(3squared)squared times (10cubed)squared'' is,
8.1 × 10⁶
We have,
Expression is,
= (3squared)squared times (10cubed)squared
It can be written as,
(3squared)squared times (10cubed)squared
(3²)² × (10³)²
9² × 1000²
81 × 1000000
8,10,00,000
8.1 × 10⁶
Thus, The value of ''(3squared)squared times (10cubed)squared'' is,
8.1 × 10⁶
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A survey asked college students the number of week days they go to a live class on campus, # of days in class (x) 0 1 2 3 4 5 (P(x) 0.34 0.22 0.22 0.10 0.08 0.04 What is the probability a student attends a live class at least 2 days a week? What is the probability a student attends a live class less than 2 days a week?
The probability that a student attends a live class at least 2 days a week is 0.58 and the probability that a student attends a live class less than 2 days a week is 0.56.
To find the probability that a student attends a live class at least 2 days a week, we need to add up the probabilities of attending class for 2, 3, 4, and 5 days. This is because attending 0 or 1 day a week means attending less than 2 days, so we need to exclude those probabilities.
P(attending at least 2 days) = P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5)
= 0.22 + 0.22 + 0.10 + 0.04
= 0.58
Therefore, the probability that a student attends a live class at least 2 days a week is 0.58.
To find the probability that a student attends a live class less than 2 days a week, we need to add up the probabilities of attending 0 or 1 day a week.
P(attending less than 2 days) = P(x = 0) + P(x = 1)
= 0.34 + 0.22
= 0.56
Therefore, the probability that a student attends a live class less than 2 days a week is 0.56.
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Assume that a study of 300 randomly selected school bus routes showed that 274 arrived on time. is it unusual for a school bus to arrive late?
Answer the question in the picture below.
Answer:
i believe it is D. because you need multiple sets of data. :)
given that z is a standard normal random variable, find c for each situation. (a) p(z < c) = 0:2119 (b) p(-c < z < -c) = 0:9030 (c) p(z < c) = 0:9948 (d) p(z > c) = 0:6915
The value of c for each situation is as follows: (a) c = -0.80 (b) c = 1.64 (c) c = 2.55 (d) c = -0.50.
We will use the z-table to find the corresponding z-scores.
(a) For p(z < c) = 0.2119, look for 0.2119 in the z-table, and find the closest value.
In this case, it is approximately 0.2118, which corresponds to a z-score of -0.80.
So, c = -0.80.
(b) For p(-c < z < c) = 0.9030, first we need to find p(z < c) since it is symmetrical around the mean.
This means p(z < c) = 1 - (1 - 0.9030) / 2 = 0.9515.
Look for 0.9515 in the z-table, and the closest value is 0.9517, which corresponds to a z-score of 1.64.
So, c = 1.64.
(c) For p(z < c) = 0.9948, look for 0.9948 in the z-table, and find the closest value.
In this case, it is approximately 0.9949, which corresponds to a z-score of 2.55.
So, c = 2.55.
(d) For p(z > c) = 0.6915, we need to find p(z < c) first.
p(z < c) = 1 - 0.6915 = 0.3085.
Look for 0.3085 in the z-table, and the closest value is 0.3085, which corresponds to a z-score of -0.50.
So, c = -0.50.
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How many numbers did Bill write? Bill wrote all the natural numbers between 9 and v (where v is greater than 9)
The number of numbers that Bill wrote (v-8) numbers.
How to form inequalities?
A mathematical phrase that states the order relationship between two integers or algebraic expressions as greater than, greater than or equal to, less than, or less than or equal to.
If Bill wrote all the natural numbers between 9 and v, then the number of numbers he wrote would be equal to the difference between v and 9 plus one (since he included both 9 and v).
v > 9
(where v is greater than 9 i.e., (v-9)+1. )
=(v-9) + 1
= v - 9 + 1
= v - 8
Therefore, the number of numbers he wrote would be (v - 8) numbers.
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What is the circumference of a circle with a diameter of 50 units? Use π = 3.14 and round your answer to the nearest hundredth.
Here are the step-by-step workings:
Circumference = π * Diameter
Circumference = 3.14 * 50 units
Circumference = 157 units
Rounded to the nearest hundredth:
Circumference = 157.00 units
Answer:
ask your teacher
What is the circumference of a circle with a diameter of 50 units? Use π = 3.14 and round your answer to the nearest hundredth
Bookwork code: L64 Calculator allowed a) What is the circumference of the shaded face? b) What is the width, w, of the rectangle? Give each answer to 1 d.p. 5 mm (0 W O Not drawn accurately
The circumference of the shaded face is 62.8 mm while the width of the rectangle is 62.8 mm
How to solve an equation?An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.
The area of a figure is the amount of space it occupies in its two dimensional state.
The radius (r) of the circle is 10mm. Hence:
Circumference = 2π * radius = 2π * 10 = 62.8 mm
The width of the rectangle = Circumference of circle = 62.8 mm
The circumference of the shaded face is 62.8 mm
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Determine zα for the following:
a.α = .0055 b.α = .09
c.α = .663
a) The value of zα for α = .0055 is -2.62.
b) The value of zα for α = .09 is 1.34.
c) The value of zα for α = .663 is .39 .
In statistics, the letter "z" usually refers to the z-score, which is a measure of how many standard deviations a particular value is from the mean. In order to determine the value of zα, we need to use the standard normal distribution table or a calculator that has this function built-in.
a. To find zα for α = .0055, we need to locate the area of .0055 in the body of the standard normal distribution table. This area is between z-scores of -2.61 and -2.62. Therefore, zα = -2.62.
b. For α = .09, we need to locate the area of .09 in the body of the standard normal distribution table. This area is between z-scores of 1.34 and 1.35. Therefore, zα = 1.34.
c. Finally, for α = .663, we need to locate the area of .663 in the body of the standard normal distribution table. This area is between z-scores of .38 and .39. Therefore, zα = .39.
In summary, zα is the z-score that corresponds to a given value of α (the level of significance). We can find this value using a standard normal distribution table or a calculator.
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-7x - 2 ≥ -3(x -6) need help
use partial fractions to find the power series of the function for 3/((x-2)(x 1))
The power series of the function 3/((x-2)(x+1)) is:
-3/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³- 1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ + ...
How to find the power series?To find the power series of the function 3/((x-2)(x+1)), we first need to find the partial fraction decomposition of the function:
3/((x-2)(x+1)) = A/(x-2) + B/(x+1)
To solve for A and B, we need to find a common denominator on the right-hand side:
3 = A(x+1) + B(x-2)
Setting x = 2, we get:
3 = A(3)
A = 1
Setting x = -1, we get:
3 = B(-3)
B = -1
Therefore, we have:
3/((x-2)(x+1)) = 1/(x-2) - 1/(x+1)
Now we can use the formula for the geometric series:
1/(1 - t) = 1 + t + t²+ t³ + ...
to write the power series of each term in the partial fraction decomposition. Substituting t = x-2 for the first term and t = -x-1 for the second term, we get:
1/(x-2) = -1/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³ + ...
1/(x+1) = -1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ - ...
Combining the two series, we have:
3/((x-2)(x+1)) = -3/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³ - 1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ + ...
Therefore, the power series of the function 3/((x-2)(x+1)) is:
-3/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³- 1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ + ...
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let c the curve be parametrized by ()=⟨2−1,−22,4−6⟩.by r(t)=⟨t2−1,t−2t2,4−6t⟩. evaluate ()r(t) at =0,t=0, =1,t=1, and =4.
Therefore, () r(t) = 4 - 8t evaluated at [tex]t=0[/tex] is 4, at [tex]t=1[/tex] is -4, and at [tex]t=4[/tex]is -28.
To evaluate the dot product ()r(t), we first need to find the coordinates of the vector :
() = ⟨2, -2, 4⟩
Then we can substitute the coordinates of r(t) into the dot product formula:
[tex]()r(t) = (2t^2 - 2 - 2t^2, -2t^2 - 2t^3, 4 - 6t) ⋅ ⟨2, -2, 4⟩[/tex]
Simplifying this expression yields:
[tex]()r(t) = 4 - 8t[/tex]
To evaluate () r(t) at different values of t, we substitute those values into the expression we just derived:
[tex]() r(0) = 4 - 8(0) = 4[/tex]
[tex]() r(1) = 4 - 8(1) = -4[/tex]
[tex]() r(4) = 4 - 8(4) = -28[/tex]
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URGENT!! Will give brainliest :)
Question 4 of 25
Describe the shape of the distribution.
A. It is uniform.
B. It is skewed.
C. It is symmetric.
D. It is bimodal.
The shape of the distribution of the box plot is described as: B: It is Skewed
What is the shape of the distribution?The box plot shape is used to indicate if a statistical particular set is either normally distributed or skewed.
There is a property of the box plot i.e. When the median is in the center of the box, and the whiskers are about the same on both flanks of the box, then the distribution is symmetric. However, when the median is anywhere to the box except the center, and if the whisker is more concise on the left or right end of the box, then the distribution is skewed.
In this question, we can clearly see that the whisker is more concise on the right end of the box, and as such we can say that the distribution is skewed.
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a sort of o(nlogn) is always preferable to a sort of o(n 2). true false
The statement " a sort of o(nlogn) is always preferable to a sort of o(n 2)" is true because sorting algorithms with O(n log n) time complexity have a lower rate of growth and are generally more efficient than sorting algorithms with O(n^2) time complexity
In general, it is true that a sorting algorithm with a time complexity of O(n log n) is preferable to a sorting algorithm with a time complexity of O(n^2), assuming other factors such as memory usage and stability are comparable.
This is because the time complexity of an algorithm describes the rate at which the algorithm's running time increases as the input size grows. In the case of sorting, O(n log n) algorithms, such as merge sort or quicksort, have a much lower rate of growth than O(n^2) algorithms, such as bubble sort or insertion sort.
This means that as the input size grows larger, the time required to sort the input using an O(n^2) algorithm can become prohibitively long, while an O(n log n) algorithm can still be practical.
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Suppose a sample of 30 MCC students is given an IQ test and the sample is found to have a standard deviation of 12.23 points. To find a 90% confidence interval for the population standard deviation:
a) Find the left-hand critical value.
b) Find the right-hand critical value.
c) Construct a 90% confidence interval for the population standard deviation.
(a) left-hand critical value is 17.71, (b) the right-hand critical value is 46.98 and (c) the 90% confidence interval for the population standard deviation is: 9.58 ≤ σ ≤ 17.45.
a) To find the left-hand critical value for a 90% confidence interval, we need to look up the corresponding value in the chi-squared distribution table with n-1 degrees of freedom, where n is the sample size. In this case, n = 30, so we look up the value with 29 degrees of freedom. The left-hand critical value is the value in the table that corresponds to the area to the left of the confidence level, which is 0.05 for a 90% confidence level. From the table, we find that the left-hand critical value is 17.71.b) To find the right-hand critical value, we use the same approach as in part (a), but this time we look up the value that corresponds to the area to the right of the confidence level. Since we want a 90% confidence level, the area to the right is also 0.05. From the table, we find that the right-hand critical value is 46.98.c) To construct the 90% confidence interval for the population standard deviation, we use the formula:lower limit ≤ σ ≤ upper limitwhere lower limit and upper limit are calculated as follows:lower limit = √((n - 1)S² / χ²_(α/2,n-1))upper limit = √((n - 1)S² / χ²_(1-α/2,n-1))where n is the sample size, S is the sample standard deviation, χ²_(α/2,n-1) is the left-hand critical value, and χ²_(1-α/2,n-1) is the right-hand critical value.Plugging in the values we found in parts (a) and (b), we get:lower limit = √((30 - 1)12.23² / 17.71) ≈ 9.58upper limit = √((30 - 1)12.23² / 46.98) ≈ 17.45Therefore, the 90% confidence interval for the population standard deviation is: 9.58 ≤ σ ≤ 17.45.For more such question on critical value
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select the function that has a well-defined inverse. group of answer choices f:→+f(x)=|x| f:→f(x)=x+4 f:→f(x)=⌈x/2⌉ f:→f(x)=2x−5
The required answer is f(x) = x + 4 and f(x) = 2x - 5
The group of answer choices, both f(x) = x + 4 and f(x) = 2x - 5 have well-defined inverses as they are both one-to-one and onto functions.
To select the function that has a well-defined inverse from the group of answer choices, we need to look for the function that satisfies the horizontal line test. The horizontal line test states that a function has a well-defined inverse if no horizontal line intersects the graph of the function more than once.
The company raised a $6 million Series A funding in 2016, led by Crosslink Capital with participation from Bertelsmann Digital Media Investments.
Out of the four answer choices, the only function that satisfies the horizontal line test is f:→f(x)=|x|. Therefore, the function f:→f(x)=|x| has a well-defined inverse.
To select the function that has a well-defined inverse, we need to identify the function that is both one-to-one and onto. Here are the given functions:
1. f(x) = |x|
2. f(x) = x + 4
3. f(x) = ⌈x/2⌉
4. f(x) = 2x - 5
the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by, [tex]f-1[/tex]
Now let's analyze each function:
1. f(x) = |x| is not one-to-one because f(1) = f(-1) = 1.
2. f(x) = x + 4 is one-to-one and onto, as every input has a unique output and every output can be achieved by a unique input.
3. f(x) = ⌈x/2⌉ is not one-to-one because f(1) = f(2) = 1.
4. f(x) = 2x - 5 is one-to-one and onto, as every input has a unique output and every output can be achieved by a unique input.
the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers.
Among the group of answer choices, both f(x) = x + 4 and f(x) = 2x - 5 have well-defined inverses as they are both one-to-one and onto functions.
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Find the sum of an arithmetic series written as Σ 20 k = 1 (− 3 k +2)
(20 on top and k=1 on the bottom of Σ )
the mean of all the sample means obtained from all random samples of a certain sample size in a sampling distribution is an example of a biased statistic. true or false
The mean of all the sample means obtained from all random samples of a certain sample size in a sampling distribution is an example of a biased statistic is False.
sampling distribution:
The mean of all the sample means obtained from all random samples of a certain sample size in a sampling distribution is an example of an unbiased statistic. This is because the sampling distribution of the mean is centered at the true population mean, and the mean of all sample means provides an estimate of that true population mean without any systematic over- or under-estimation.
However, individual sample means can be biased if there are any issues with the sampling process or if the sample is not representative of the population.
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calculate the final thickness of the silicon dioxide on a wafer
The final thickness of the silicon dioxide on a wafer is given by: Initial thickness + (growth rate x oxidation time)
To calculate the final thickness of the silicon dioxide on a wafer, you will need to know the initial thickness of the oxide layer and the duration of the oxidation process.
The growth rate of silicon dioxide is dependent on temperature and can be determined from the literature. Once you have this information, you can use the following formula to calculate the final thickness:
Final thickness = initial thickness + (growth rate x oxidation time)
It is important to note that the final thickness may be affected by any post-oxidation processing steps, such as etching or cleaning, that may remove some of the oxide layer.
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List the prime factors of 50
Answer:
[tex]2[/tex] × [tex]5^{2}[/tex]
Step-by-step explanation:
2 X 5 X 5 = 50
Hope this helps
the exponential mode a=979e 0.0008t describes the population,a, of a country in millions, t years after 2003. use the model to determine the population of the country in 2003
The population of the country in 2003 was 979 million.
We are given that;
a=979e 0.0008t
Now,
To find the population of the country in 2003, we need to plug in t = 0 into the model, since 2003 is the starting year.
a = 979e^(0.0008t)
a = 979e^(0.0008(0))
a = 979e^0
a = 979(1)
a = 979
Therefore, by the exponential mode the answer will be 979 million.
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The population of the country in 2003 was 979 million.
We are given that;
a=979e 0.0008t
Now,
To find the population of the country in 2003, we need to plug in t = 0 into the model, since 2003 is the starting year.
a = 979e^(0.0008t)
a = 979e^(0.0008(0))
a = 979e^0
a = 979(1)
a = 979
Therefore, by the exponential mode the answer will be 979 million.
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suppose that AB is invertible then (AB)^−1 exists. We also know (AB)^−1=B^−1A^−1. If we let C=(B^−1A−^1A) then by the invertible matrix theorem we see that since CA=I(left inverse) then B is invertible. Would this be correct?
The invertible (AB)^-1 exists and is equal to B^-1A^-1. Yes, that is correct.
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melinda needed to mail a package. she used $0.02 stamps and $0.10 stamps to mail package. if she used 15 stamps worht $.78 how many $0.10 stamps did she use
Therefore, Melinda used 6 $0.10 stamps in the given equation.
Let's say Melinda used x $0.02 stamps and y $0.10 stamps.
From the problem, we know that:
x + y = 15 (the total number of stamps used is 15)
0.02x + 0.1y = 0.78 (the total value of the stamps used is $0.78)
To solve for y, we can use the first equation to solve for x:
x = 15 - y
Substituting into the second equation:
0.02(15 - y) + 0.1y = 0.78
Expanding and simplifying:
0.3 - 0.02y + 0.1y = 0.78
0.08y = 0.48
y = 6
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What is the order of 8 + 12Z in the factor group Z/12Z?
The order of 8 + 12Z in the factor group Z/12Z is 3.
This is because the order of an element a in a group G is the smallest positive integer n such that aⁿ = e, where e is the identity element of G. In this case, (8 + 12Z)³ = 8³ + 12Z = 8 + 12Z = 0 + 12Z, which is the identity element of Z/12Z. Therefore, the order of 8 + 12Z is 3.
To find the order of an element in a factor group, we first need to determine the cosets of the group modulo the subgroup. In this case, we have Z/12Z, which is the integers modulo 12. The subgroup is 12Z, which consists of all multiples of 12.
We can write the cosets of 12Z as {0 + 12Z, 1 + 12Z, 2 + 12Z, ..., 11 + 12Z}. Each of these cosets contains an element that is congruent to 8 modulo 12. For example, 8 + 12Z is in the coset 8 + 12Z, and 20 + 12Z is in the coset 8 + 12Z.
To find the order of 8 + 12Z, we need to find the smallest positive integer n such that (8 + 12Z)ⁿ is equal to the identity element of Z/12Z, which is 0 + 12Z. We can compute (8 + 12Z)² as (8 + 12Z)(8 + 12Z) = 64 + 96Z = 4 + 12Z, since 64 is congruent to 4 modulo 12 and 96 is a multiple of 12. Therefore, (8 + 12Z)² is not equal to the identity element.
Next, we compute (8 + 12Z)³ as (8 + 12Z)(8 + 12Z)(8 + 12Z) = 512 + 864Z = 8 + 12Z, since 512 is congruent to 8 modulo 12 and 864 is a multiple of 12. Therefore, (8 + 12Z)³ is equal to the identity element, and the order of 8 + 12Z is 3.
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