4. in a survey of 3611 adult americans 18 years and older conducted in july 2010, it was found that 542 have used their smartphone to make a purchase. find the 90% confidence interval for the population proportion of adult americans who have used their smartphone to make a purchase. interpret this confidence interval in a sentence.

The equivalent expression is  log(6) + log(7)

There are two rules for logarithmic relations that we need to know here, these are:

log(a*b) = log(a)+ log(b)

log(a/b) = log(a) - log(b)

Here we have the product of 6 and 7 in the argument, then we can write the equivalent expression:

log(6*7) = log(6) + log(7)

That is the correct option.

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The Type II error in this situation is the fourth option: (d) based on the sample mean, the owner concludes that the mean weight of all of the bunches of bananas is not less than 3.54 pounds when the true mean weight is actually less than 3.54 pounds.

A Type II error occurs when the null hypothesis (in this case, that the mean weight of the banana bunches is 3.54 pounds) is not rejected, even though it is false. In other words, the owner fails to reject the shipment of bananas, even though it does not meet the required weight standard. In this scenario, the owner may face a loss in business or reputation due to the low quality of bananas, and may also incur losses by selling the underweight bananas at a lower price or even disposing of them.

The probability of making a Type II error can be minimized by increasing the sample size or decreasing the significance level (alpha level) of the test. However, a Type II error can never be completely eliminated.

Therefore, the correct option is d.

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Answer: To find the initial population size of the fish species, we can simply substitute t = 0 into the exponential function and evaluate:

P(0) = 1000 / (1 + 7e^(0.3*0)) = 1000 / (1 + 7e^0) = 1000 / 8 = 125

Therefore, the initial population size of the fish species was approximately 125 individuals.

To find the population size after 7 years, we can substitute t = 7 into the exponential function and evaluate:

P(7) = 1000 / (1 + 7e^(0.3*7)) ≈ 638

Therefore, the population size of the fish species after 7 years was approximately 638 individuals (rounded to the nearest whole number).

The initiall population size is: 125. Rounded to the nearest whole number, the population size after 7 years is 18.

Initial population size is:
To find the initial population size, we need to find P(0). Plug t=0 into the given equation:

P(0) = 1000/(1+7e^(0.3*0))
P(0) = 1000/(1+7*1) = 1000/8 ≈ 125

The initial population size of the species is approximately 125.

Population size after 7 years is:
To find the population size after 7 years, we need to find P(7). Plug t=7 into the given equation:

P(7) = 1000/(1+7e^(0.3*7))
P(7) ≈ 1000/(1+7e^2.1) ≈ 1000/(1+7*8.166) ≈ 1000/(1+57.162) ≈ 1000/58.162 ≈ 17.19

The population size after 7 years is approximately 17.

Your answer:
The initial population size is: 125

Population size after 7 years is: 17

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The unknown side length in triangle B has a measure of 7.5 units.

It is given that Alejandro reduced triangle A proportionally.

It means triangle A and B are similar and their corresponding sides are proportional.

Scale factor = 6/12

=1/2

Each side of triangle A is changed by a factor of 1/2.

Let the unknown side of triangle B be x.

x/15=1/2

2x=15

x=7.5

Therefore, the unknown side length in triangle B has a measure of 7.5 units.

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The required, 4.5 × 10⁵ as an ordinary number is 450,000.

An ordinary number is a number that is expressed in the usual way, using digits 0-9 without any exponent notation or other mathematical symbols.

4.5 × 10⁵ means 4.5 multiplied by 10 raised to the power of 5. To write this as an ordinary number, we simply need to perform this multiplication:

4.5 × 10⁵ = 4.5 × 100,000 = 450,000

Therefore, 4.5 × 10⁵ as an ordinary number is 450,000.

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Answer: approximately 25.13 centimeters.

Step-by-step explanation: If the diameter of a circle is 8cm, then the radius is half of that, which is 4cm. We can use this information to calculate the area and circumference of the circle as follows:

Area of circle = π * r^2

= π * 4^2

= π * 16

= 16π

≈ 50.27 cm^2 (rounded to two decimal places)

Circumference of circle = 2π * r

= 2π * 4

= 8π

≈ 25.13 cm (rounded to two decimal places)

Therefore, the area of the circle is approximately 50.27 square centimeters and the circumference is approximately 25.13 centimeters.

no picture is not right

1. The equation y + 3/4(y+30) = 478 can be used to find y, Elena's score in her first game. False

2. The difference between the score in Elena's first game and her second game is 34. True

3. The equation z/4y + 30 = 478 can be used to find y, Elena's score in her first game. False

4.  Elena's scores 222 in her first game

To find Elena's scores in the game, we used the equation:

x = 3/4y + 30 since x + y = 478

3/4y + 30 + y = 478

7/4y + 30 = 478

7/4y = 478 - 30 = 448

448 x 4 /7 = 256

It means that Elena's scored 256 in her first game and 222 in her second game.

The above answer is based on the questions below as seen in the picture

Elena bowls two games on Saturday. Her serve in the second game is 30 more than 3/4 of her score in the first game. Elena's total score for the two games is 478.

Determine with each statement about Elena's bowling games is true;

1 The equation y + 3/4(y+30) = 478 can be used to find y, Elena's score in her first game.

2. The difference between the score in Elena's first game and her second game is 34.

3. The equation z/4y + 30 = 478 can be used to find y, Elena's score in her first game.

4. Elena's scores 222 in her first game.

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x ≈ 0.625.

To solve the exponential equation 3 + 4^(4x-3) + 6 = 11, we first need to isolate the exponential term.

Subtracting 3 and 6 from both sides, we get:

4^(4x-3) = 2

To solve for x, we can take the natural logarithm of both sides:

ln(4^(4x-3)) = ln(2)

Using the property of logarithms that ln(a^b) = b*ln(a), we can simplify the left side:

(4x-3)ln(4) = ln(2)

Dividing both sides by ln(4), we get:

4x-3 = ln(2)/ln(4)

Simplifying the right side using a calculator, we get:

4x-3 ≈ -0.5

Adding 3 to both sides, we get:

4x ≈ 2.5

Dividing by 4, we get:

x ≈ 0.625

Therefore, the exact solution with natural logarithms is:

x = (ln(2)/ln(4) + 3)/4

And the approximate solution correct to three decimal places is:

x ≈ 0.625

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The intersection change has significantly increased the number of traffic accidents. If not, you would fail to reject the null hypothesis and not have enough evidence to claim a significant increase in accidents due to the intersection change.

A) The Null Hypothesis (H0) is that the intersection change has not significantly increased the number of traffic accidents.

B) Without specific data, I cannot calculate the exact Test Statistic value. You'll need to perform a hypothesis test using the appropriate formula for your data set.

C) The Critical Value also depends on your data and the level of significance (alpha) you choose, typically 0.05 or 0.01. Use a statistical table or software to find the critical value based on your chosen level of significance and the degrees of freedom.

D) Similar to B and C, the p-value cannot be calculated without specific data. Compare the calculated p-value to the chosen level of significance (alpha) to determine whether to reject or fail to reject the null hypothesis.

E) Based on the Test Statistic, Critical Value, and p-value, you can draw conclusions about the null hypothesis. If the Test Statistic is greater than the Critical Value or the p-value is less than the chosen level of significance (alpha), you would reject the null hypothesis.

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[tex]\textit{surface area of a sphere}\\\\ SA=4\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ SA=2826 \end{cases} \implies 2826=4\pi r^2 \\\\\\ \cfrac{2826}{4\pi }=r^2\implies \cfrac{1413}{2\pi }=r^2\implies \sqrt{\cfrac{1413}{2\pi }}=r \\\\[-0.35em] ~\dotfill[/tex]

[tex]\textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=\sqrt{\frac{1413}{2\pi }} \end{cases}\implies V=\cfrac{4\pi }{3}\cdot \left( \sqrt{\cfrac{1413}{2\pi }} \right)^3 \\\\\\ V=\cfrac{4\pi }{3}\cdot \cfrac{4239\sqrt{157}}{2\pi \sqrt{2\pi }}\implies V=\cfrac{2826\sqrt{157}}{\sqrt{2\pi }}\implies V\approx 14126.42~mm^3[/tex]

Based on the information provided, the researcher's hypothesis is that people who use their phones before bed get less sleep than the recommended 7 hours per night by the national sleep foundation.

The researcher samples 30 people who report using their phones immediately before bedtime and records their average hours of sleep per night. To test the hypothesis, a one-sample t-test is conducted to compare the number of hours of sleep the phone users get relative to the national sleep foundation recommendations.

The results of the t-test will indicate whether the average number of hours of sleep the phone users get is significantly different from the recommended 7 hours per night. If the results show that the phone users get significantly less sleep than the recommended amount, this could indicate a negative effect of phone use on sleep quality.
The researcher hypothesizes that people who use their phones before bed get less sleep than the National Sleep Foundation's recommendation of at least 7 hours per night for adults. A one-sample t-test is conducted to compare the average number of hours of sleep for a sample of 30 phone users against this recommended value. The test will help determine if there's a significant difference between the sleep duration of phone users and the recommended sleep duration.

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The Kanizsa triangle illusion helps us understand that our perceptual experiences are not simply the sum of the individual sensory inputs, but rather the result of a complex and variable process of perceptual organization.

Gestalt is a German word meaning "shape" or "form," and in psychology, it refers to a set of principles that describe how people perceive and organize sensory information into meaningful wholes. These principles propose that the whole is greater than the sum of its parts, and that we tend to organize our perceptual experiences into coherent, holistic forms rather than isolated, unrelated sensations.

Experimental examples such as the Necker cube, visual cliff, and others help demonstrate principles of perceptual organization by highlighting how our minds naturally try to impose structure and order on sensory input. For example, the Necker cube is a two-dimensional drawing that can be perceived as a cube that can be viewed from different angles. However, as one stares at the image, it appears to flip back and forth between different possible interpretations. This phenomenon illustrates the Gestalt principle of figure-ground, which describes how we tend to perceive objects as being distinct from their surrounding context.

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I disаgree with thе stаtеmеnt becаuse thе surfасe аreа аnd volumе оf а cylindеr аre indеpеndеnt оf eаch othеr аnd cаn vаry indеpеndеntly.

An exаmple: twо сylinders cаn hаve thе sаme volumе but different heights or rаdii, which would rеsult in different surfасe аrеаs. Convеrsеly, twо сylinders cаn hаve thе sаme surfасe аreа but different heights or rаdii, which would rеsult in different volumеs.

To demonstrаte this, you cаn perfоrm аn exрeriment by tаking twо сylinders оf thе sаme volumе but different rаdii аnd heights. Fill eаch cylindеr with wаter to thе sаme lеvеl, аnd mаrk this lеvеl on thе insidе оf thе cylindеr. Then рour thе wаter from one cylindеr into thе othеr аnd mаrk thе nеw lеvеl.

Neхt, meаsure thе heights оf thе wаter lеvеls in both сylinders аnd cаlculаte thе surfасe аreа оf eаch cylindеr using thе formulа for thе surfасe аreа оf а cylindеr:

Surfаce аreа = 2πrh + 2πr^2

Уou will find thаt thе twо сylinders hаve different surfасe аrеаs, even though thеy hаve thе sаme volumе. This exрeriment demonstrаtes thаt thе stаtеmеnt "If thе volumе оf twо сylinders аre equаl, thеn thеir surfасe аrеаs must be equаl" is fаlse.

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The coordinate of D’ after the dilation is (-6, 0)

From the question, we have the following parameters that can be used in our computation:

D = (-2, 0)

Scale factor = 3

The coordinate of D’ after the dilation is calculated as

D' = D * Scale factor

Substitute the known values in the above equation, so, we have the following representation

D' = (-2, 0) * 3

Evaluate

D' = (-6, 0)

Hence, the image is (-6, 0)

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The solution for the given quadratic equation v² - 9v = -4 are v = [9 ± √65] / 2

To solve a quadratic equation in the form of ax² + bx + c = 0 using the quadratic formula, we can use the following formula:

x = [-b ± √(b² - 4ac)] / 2a

In this case, we have the equation v² - 9v = -4, which can be rearranged to the standard form of ax² + bx + c = 0 by adding 4 to both sides:

v² - 9v + 4 = 0

Now we can identify the values of a, b, and c:

a = 1, b = -9, c = 4

Substituting these values into the quadratic formula, we get:

v = [9 ± √(81 - 16)] / 2

Simplifying under the square root:

v = [9 ± √65] / 2

These are the two solutions for v, which can be expressed as decimals rounded to the nearest hundredth or as exact fractions.

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According to a report by the U.S. Department of Agriculture, in 2019, 7.7% of head-of-household Americans were uncertain of having enough food to meet the needs of all the members of their household.

This is a slight increase from the previous year. It is important to note that food insecurity disproportionately affects certain groups, such as households with children, low-income households, and households headed by minorities. The COVID-19 pandemic has also had a significant impact on food insecurity rates in the United States.

In 2019, approximately 10.5% of head-of-household Americans were uncertain of having enough food to meet the needs of all members of their household. This statistic, also known as food insecurity, reflects the number of households that had difficulty at some point during the year providing enough food for all their members due to a lack of resources. It is essential to address this issue to ensure that all families have access to adequate nutrition and can lead healthy lives.

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The side lengths which could form right triangles using Pythagoras theorem are:

6, 8, 10

8, 15, 17

12, 35, 37

Given are certain options which can form right triangle.

By Pythagoras theorem,

Square of longest side = sum of the squares of the shortest side

3, 4, 7

3² + 4² = 9 + 16 = 25 ≠ 49

This is not a right triangle.

5, 8, 11

5² + 8² = 25 + 64 = 89 ≠ 121

This is not a right triangle

6, 8, 10

6² + 8² = 36 + 64 = 100 = 10²

This is a right triangle.

8, 15, 17

8² + 15² = 64 + 225 = 289 = 17²

This is a right triangle

12, 35, 37

12² + 35² = 144 + 1225 = 1369 = 37²

This is also a right triangle.

Hence the right triangles are of side lengths :

6, 8, 10

8, 15, 17

12, 35, 37

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Step-by-step explanation:

in geometry, a minor arc is an arc that measures less than 180°, and a major arc is one that measures greater than 180°. Arc measure is the angle measure of the center angle corresponding to the arc. An arc is a part of the circumference of a circle.

The probability of 12 successes in 30 trials with a 0.20 probability of success on a single trial is approximately 0.0139.

To find the probability of x successes in n trials with a probability p of success on a single trial, you can use the binomial probability formula:
P(x) = C(n, x) * (p^x) * ((1-p)^(n-x))
where C(n, x) is the number of combinations of n items taken x at a time.
In this case, n = 30, x = 12, and p = 0.20. Plug these values into the formula:
P(12) = C(30, 12) * (0.20^12) * ((1-0.20)^(30-12))
Calculate C(30, 12), which is the number of combinations of 30 items taken 12 at a time:
C(30, 12) = 30! / (12! * (30-12)!) = 86493225
Now, calculate the rest of the equation:
P(12) = 86493225 * (0.20^12) * (0.80^18) ≈ 0.0139

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Answer:

  y = -3√(x -1) +2

Step-by-step explanation:

You want the equation of the translated, reflected, and scaled square root function in the given graph.

We can find the amount of translation by looking at the position of the point that corresponds to (0, 0) on the graph of the parent function. That end point where the slope is vertical is located at (1, 2) on the given graph.

When a function y = f(x) is translated by (h, k), it becomes ...

  y = f(x -h) +k

For (h, k) = (1, 2) the translated function is ...

  y = f(x -1) +2 = √(x -1) +2

When the function is scaled, its value is multiplied by the scale factor. If that factor is 'a', our scaled and translated function is ...

  y = a√(x -1) +2

To find the scale factor, we can use the second point on the curve: (2, -1). Using these values for x and y gives ...

  -1 = a√(2 -1) +2

  -3 = a . . . . . . . . . . subtract 2, simplify

The transformed function is ...

  y = -3√(x -1) +2

__

Additional comment

The function is scaled before it is translated, so the scale factor does not apply to the translation.

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If both eigenvalues of a 2x2 matrix A are complex with nonzero real part, then the critical point of the system x' = Ax is a center.

A center is a type of critical point where the solutions of the system oscillate around the critical point, without converging or diverging. The center has the property that the solutions move along closed trajectories, which are ellipses in the case of a 2x2 system. The orientation and size of the ellipses depend on the values of the eigenvalues and eigenvectors of the matrix A.

In general, centers are not stable or unstable in the sense of Lyapunov. Instead, they are neutral points where the solutions of the system do not change in magnitude, but only in direction.

The classification of a critical point as a center is important because it indicates the existence of periodic solutions in the system. These periodic solutions are of interest in many applications, such as in the study of oscillatory behavior in physical systems, or in the analysis of biological rhythms.

In summary, if both eigenvalues of a 2x2 matrix A are complex with nonzero real part, then the critical point of the system x' = Ax is a center, and the solutions of the system move along closed trajectories, without converging or diverging.

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[tex]-\cos^2(x)+\sin(x)=1\implies -[1-\sin^2(x)]+\sin(x)=1 \\\\\\ \sin^2(x)+\sin(x)-2=0\implies [\sin(x)]^2+\sin(x)-2=0 \\\\\\ ( ~~ \sin(x)-1 ~~ )( ~~ \sin(x)+2 ~~ )=0 \\\\[-0.35em] ~\dotfill\\\\ \sin(x)-1=0\implies \sin(x)=1\implies x=\sin^{-1}(1)\implies x=\frac{\pi }{2}[/tex]

now, what's wrong with the 2nd factor? sin(x) + 2 = 0?

well, we can go ahead and make it sin(x) = -2, however, let's recall that sine is never less than -1 or even more than 1, so that's out of range for sine.

In this probability experiment, the sample space is S = {2,3,4,5,6,7,8,9,10,11,12,13). The event E is defined as E = {3,4,5,6). Assuming that each outcome is equally likely, we can list all the outcomes of the experiment as follows:

1. If we roll a 2, it is not included in event E, so the outcome is not included.
2. If we roll a 3, it is included in event E, so the outcome is included.
3. If we roll a 4, it is included in event E, so the outcome is included.
4. If we roll a 5, it is included in event E, so the outcome is included.
5. If we roll a 6, it is included in event E, so the outcome is included.
6. If we roll a 7, it is not included in event E, so the outcome is not included.
7. If we roll an 8, it is not included in event E, so the outcome is not included.
8. If we roll a 9, it is not included in event E, so the outcome is not included.
9. If we roll a 10, it is not included in event E, so the outcome is not included.
10. If we roll an 11, it is not included in event E, so the outcome is not included.
11. If we roll a 12, it is not included in event E, so the outcome is not included.
12. If we roll a 13, it is not included in event E, so the outcome is not included.

Therefore, the outcomes of the experiment that are included in event E are 3, 4, 5, and 6.

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This indicates that there is no clear direction in which the system moves, and higher-order terms in the Taylor series may cause it to move away from the origin.

Theorem 9.6.2 provides conditions for instability when the origin is an unstable critical point. The statement of the theorem is as follows:

Suppose that the system of differential equations given by

dx/dt = f(x,y)

dy/dt = g(x,y)

Has an unstable critical point at the origin, (0,0). That is, the origin is a critical point, but it is not stable. Then the following conditions must hold for the system to be unstable at the origin:

At least one eigenvalue of the Jacobian matrix evaluated at the origin is positive.

The eigenvector corresponding to the positive eigenvalue points away from the origin.

Alternatively, if the Jacobian matrix evaluated at the origin has a repeated eigenvalue of zero, then the system may also be unstable. In this case, we need to examine the higher-order terms in the Taylor series expansion of the system near the origin to determine its stability.

Intuitively, the conditions for instability tell us that if there is any direction in which the system moves away from the origin, then the origin is unstable. This can happen if the eigenvalues of the Jacobian matrix have a positive real part, which causes the system to move away from the origin in that direction. It can also happen if the Jacobian matrix has a repeated eigenvalue of zero, since this indicates that there is no clear direction in which the system moves, and higher-order terms in the Taylor series may cause it to move away from the origin.

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Option D (9) is not a probability and is not a possible answer choice for a probability question.

To find the probability of the intersection of events A and B (denoted as P(A ∩ B)), we need to know how many elements are in both A and B. Since there are 2 repeated elements between the events, this means that there are a total of 7 - 2 = 5 distinct elements between them.

Therefore, P(A ∩ B) = 5/7.

Note that none of the answer choices provided match this result exactly. Option A is the closest with a probability of 4/7, but this is the probability of A alone, not the intersection of A and B. Option B (2/7) is the probability of B alone and option C (5/7) is the probability of either A or B occurring, not the intersection.

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The product of the two matrices is  [2 - 3]    x  [1  -  4]

                                                            [ 3   2]       [4     1]

option B.

A product matrix is also known as matrix multiplication, which involves the multiplication of two matrices, to get or simply to a single matrix.

For example, if P and Q are the two matrices, then the product of the two matrices P and Q are denoted by:

Y = PQ.

For the given question; we have the first matrix as; (2 - 3i) and the second matrix as (1 - 4i).

Matrix (2 - 3i) is transformed to [2 - 3]

                                                   [ 3   2]

Matrix (1 - 4i) is transformed to [1  -  4]

                                                  [4     1]

The product of the two matrices =  [2 - 3]    x  [1  -  4]

                                                          [ 3   2]       [4     1]

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The expressions below best represent within the context of the word problem is x + (x - 24) = 68, Sarah's age is 46 years. Mother's age is 46 - 24 = 22. Therefore, Sarah is 46 years old, and her mother is 22 years older than her, making her mother 68 years old.

In the given word problem, we are given two pieces of information about Sarah and her mother's ages. Firstly, we know that Sarah is 24 years younger than her mother. Secondly, we know that the sum of their ages is 68. To find out Sarah's age, we need to represent it using an expression. Let's say Sarah's age is 'x'. Therefore, the expression that best represents Sarah's age is 'x'.

Now, we also need to represent Sarah's mother's age using an expression. As we know that Sarah's mother is 24 years older than her, we can subtract 24 from Sarah's age to get her mother's age. So, the expression that best represents Sarah's mother's age is 'x - 24'.

To find Sarah's age, we can use the sum of their ages, which is 68. We know that Sarah's age is 'x', and her mother's age is 'x - 24'. Therefore, we can write an equation:

x + (x - 24) = 68

Solving this equation, we get:

2x - 24 = 68

2x = 92

x = 46

Hence, Sarah is 46 years old.

In the given word problem, we are asked to find the age of Sarah, knowing that she is 24 years younger than her mother and the sum of their ages is 68. We can use the expressions x and x - 24 to represent Sarah's and her mother's ages, respectively.

Let x represent Sarah's age. Since Sarah is 24 years younger than her mother, we can represent her mother's age as x - 24. According to the problem, the sum of their ages is 68. We can now set up an equation using this information:

x (Sarah's age) + (x - 24) (Mother's age) = 68

Solve the equation to find the value of x, which represents Sarah's age:

x + x - 24 = 68
2x - 24 = 68

Now, add 24 to both sides of the equation:

2x = 92

Next, divide both sides by 2:

x = 46

So, Sarah's age is 46 years. To find her mother's age, we can substitute the value of x in the expression x - 24:

Mother's age = 46 - 24 = 22

Therefore, Sarah is 46 years old, and her mother is 22 years older than her, making her mother 68 years old.

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The equation of the graph or function y=-1/9x²

From the given instruction we obtain a parabola with an x-coordinate equal to the midpoint of I of the endpoints of the base or −6+6/2 =0

And y-coordinate equal to the midpoint of the y-coordinates of the vertex and an endpoint of the base or 4+(−4)/2=0

So the vertex is (0,0)

From the function y=x² above graph is the reflection about x-axis

so we have y=−ax²

So, to find "a" we will put (6,-4) in equation 1 we get

-4=-a(6)²

a=1/9

Hence, the equation of the graph is y=-1/9x²

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If teacher is distributing sheets in a class, then the equation which is used to find number of students in class is (d) 2x+8 = 3x - 11​.

A "Linear-Equation" is a mathematical equation that represents a straight line in a coordinate plane. It is of form : y = mx + b

where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept (the point at which the line crosses the y-axis).

Let number of students in class be denotes as "x",

If each student get 2 sheets, then 8 sheets are remaining, it is mathematically represented as : 2x + 8 ,

If each student get 3 sheets each, then there would be 11 sheets less, and this is represented as : 3x - 11,

So, the equation which is used to find number of students in class is 2x+8=3x-11,

Therefore, the correct option is (d).

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The given question is incomplete, the complete question is

If a teacher were to distribute sheets of paper so that each student got two sheets, there would be 8 sheets remaining. However, if three sheets were given to each student, the teacher would be 11 sheets short. Which equation could be used to find how many students are in the class?

(a) 2(x - 8) = 3(x + 11)

(b) 2(x + 8) = 3(x - 11)

(c) 2x-8 = 3x + 11

(d) 2x+8 = 3x - 11​

Answer:

y(t) = 3/4exp(4t) - 1/4exp(-4t) - 1/2

z(t) = -3/4exp(4t) - 1/4exp(-4t) - 2

Step-by-step explanation:

Given system of differential equations:

dy/dt = 2y + 2z

dz/dt = 2y + 2z

We can write this system in matrix form as:

d/dt [y z] = [2 2] [y z]

Let A = [2 2]. Then the system can be written as:

d/dt [y z] = A[y z]

The solution to this system is given by:

[y z] = exp(At) [y(0) z(0)]

where exp(At) is the matrix exponential of At.

To find exp(At), we first need to find the eigenvalues and eigenvectors of A. The characteristic equation of A is:

det(A - lambdaI) = 0

=> det([2-lambda 2; 2 2-lambda]) = 0

=> (2-lambda)(2-lambda) - 4 = 0

=> lambda1 = 4, lambda2 = 0

The eigenvectors corresponding to lambda1 = 4 and lambda2 = 0 are:

v1 = [1 1] and v2 = [-1 1]

We can now write A as:

A = PDP^-1

where P = [v1 v2] and D = [4 0; 0 0]. Then,

exp(At) = Pexp(Dt)P^-1

We can compute exp(Dt) as:

exp(Dt) = [exp(4t) 0; 0 1]

Therefore,

exp(At) = [1/2 1/2; -1/2 1/2] [exp(4t) 0; 0 1] [1/2 -1/2; 1/2 1/2]

Now, we can find the solution to the system as:

[y z] = exp(At) [y(0) z(0)]

=> [y z] = [1/2 1/2; -1/2 1/2] [exp(4t) 0; 0 1] [1/2 -1/2; 1/2 1/2] [-1; -2]

=> [y z] = [1/2 1/2; -1/2 1/2] [exp(4t) 0; 0 1] [3/2; -1/2]

=> [y z] = [1/2 1/2; -1/2 1/2] [3/2exp(4t); -1/2]

=> [y z] = [3/2exp(4t)/2 - 1/2exp(-4t)/2; -3/2exp(4t)/2 - 1/2exp(-4t)/2]

Therefore, the solution to the system of differential equations with the given initial conditions is:

y(t) = 3/4exp(4t) - 1/4exp(-4t) - 1/2

z(t) = -3/4exp(4t) - 1/4exp(-4t) - 2