Tell whether the two quantities vary directly. Explain your reasoning.
the number of correct answers on a test and the score on the test
Choose the correct answer below.
OA. No, they do not vary directly. When one quantity increases, the other quantity does not increase.
OB. No, they do not vary directly. When one quantity increases, the other quantity also increases.
C. Yes, they vary directly. When one quantity increases, the other quantity also increases.
OD. Yes, they vary directly. When one quantity increases, the other quantity does not increase.

For values of y near 0, the functions can be ordered in increasing order as follows: 1 - cos(y) < ln(1 + y^2) < 1/(1 - y^2) - 1.

To determine the increasing order of the functions for values of y near 0 using their Taylor expansions, let's calculate the Taylor series expansions for each function and compare them.

(a) 1 - cos(y):

The Taylor series expansion for 1 - cos(y) centered at y = 0 is:

1 - cos(y) = 0 + (1/2!)y^2 + (0) + ...

The second-order term is positive, and all higher-order terms are non-negative. Therefore, for values of y near 0, the function 1 - cos(y) is increasing.

(b) ln(1 + y^2):

The Taylor series expansion for ln(1 + y^2) centered at y = 0 is:

ln(1 + y^2) = (0) + (1/1)(y - 0) + (0) + ...

The first-order term is positive, and all higher-order terms are non-negative. Therefore, for values of y near 0, the function ln(1 + y^2) is increasing.

(c) 1/(1 - y^2) - 1:

The Taylor series expansion for 1/(1 - y^2) - 1 centered at y = 0 is:

1/(1 - y^2) - 1 = 1/(1 - 0^2) - 1 + (0) + ...

The constant term is positive, and all higher-order terms are non-negative. Therefore, for values of y near 0, the function 1/(1 - y^2) - 1 is increasing.

Therefore, in increasing order for values of y near 0, we have:

1 - cos(y) < ln(1 + y^2) < 1/(1 - y^2)

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the instantaneous velocity function of the object moving along the line described by y=f(x)=6x^2-5x at time x (in seconds) is v = f'(x) = 12x - 5 feet per second.

the instantaneous velocity of an object is the rate of change of its position at a particular moment in time. In other words, it is the slope of the tangent line to the position function at that specific point.

To find the instantaneous velocity function, we take the derivative of the position function with respect to time. In this case, the derivative of f(x) is f'(x) = 12x - 5.

To find the velocity at times x = 1, 3, and 5 seconds, we simply plug in those values for x into the instantaneous velocity function. Therefore, the velocity at x = 1 second is v(1) = f'(1) = 12(1) - 5 = 7 feet per second. The velocity at x = 3 seconds is v(3) = f'(3) = 12(3) - 5 = 31 feet per second. Finally, the velocity at x = 5 seconds is v(5) = f'(5) = 12(5) - 5 = 55 feet per second.

the instantaneous velocity function of the object is v = f'(x) = 12x - 5 feet per second, and the velocity at times x = 1, 3, and 5 seconds is 7 feet per second, 31 feet per second, and 55 feet per second, respectively.

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The particular solution is y = (-9/8) * (9x + 81) e^(-x/9) + 85.125.

To solve this differential equation, we can use separation of variables.

dy/dx = -9x/8 e^(-x/9)

dy = (-9/8)x e^(-x/9) dx

Integrating both sides, we get:

y = (-9/8) * (9x + 81) e^(-x/9) + C

where C is the constant of integration.

To find the particular solution, we need to use the initial condition. Let's say that y(0) = 4.

Then, when x = 0, we have:

4 = (-9/8) * (0 + 81) e^(0) + C

C = 4 + (9/8) * 81

C = 85.125

Therefore, the particular solution is:

y = (-9/8) * (9x + 81) e^(-x/9) + 85.125

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If the rate of interest is the same in both years, the rate of interest is 12%.

The man took a loan of Rs.10,000 at simple interest. He paid Rs. 5,000 at the end of first year in which interest is also included and Rs. 6,944 at the end of the second year and clears the debt.

Here is the solution:

Interest paid in the first year = Rs. (10,000 - 5,000) = Rs. 5,000

Interest paid in the second year = Rs. (6,944 - 10,000) = Rs. -3,056

Total interest paid = Rs. (5,000 + (-3,056)) = Rs. 1,944

Principal amount = Rs. 10,000

Rate of interest = (100 * 1,944) / (10,000 * 2) = 12%

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The sample size 24 and sample proportion (p) is 0.5 will be a normal curve be used to approximate the sampling distribution

The condition for a normal curve to be used to approximate the sampling distribution is that the sample size should be large enough such that both np and n(1-p) are greater than or equal to 10.

Let's check the options one by one:

n = 24; p = 0.5

Here, np = 24 x 0.5 = 12 and

n(1-p) = 24 x 0.5 = 12

Both of which are greater than or equal to 10.

So, a normal curve can be used to approximate the sampling distribution.

n = 20, p = 0.6

n×p = 12, n×(1-p) = 8, so a normal curve cannot be used.

C. n = 24, p = 0.4: n × p = 9.6, n ×(1-p) = 14.4, so a normal curve cannot be used.

D. n = 20, p = 0.3: n × p = 6, n×(1-p) = 14, so a normal curve cannot be used.

Therefore, the sample size is 24 and sample proportion (p) is 0.5 will be a normal curve be used to approximate the sampling distribution

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(1) False. A p-value of 4 indicates that there is a high probability of obtaining the observed results by chance alone.
(2) False. The p-value is the probability of obtaining the observed results or more extreme results if the null hypothesis is true.

False. A p-value of .4 means we should not necessarily accept that the null hypothesis is true. The p-value represents the probability of obtaining a result as extreme or more extreme than the one observed, assuming the null hypothesis is true. A common threshold to reject the null hypothesis is a p-value less than 0.05. Since 0.4 is greater than 0.05, we do not have enough evidence to reject the null hypothesis, but it does not mean we should accept it as true.

False. The p-value is not the probability that the null hypothesis is true. It is the probability of observing the given results (or more extreme results) assuming the null hypothesis is true.

In the strep test scenario, the first test made a Type II error. This occurs when the test fails to reject a false null hypothesis, meaning the test showed a negative result (no strep) when the student actually had strep.

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The answers are explained in the solution below.

Given is map of transformations,

a) ΔABC → ΔA'B'C' = Translation

b) ΔABC → ΔA''B''C'' = Translation → Reflection

c) ΔABC → ΔA'''B'''C''' = Translation → Reflection → Rotation.

d) The composition of the transformation is =

T₂ ⁰ [tex]r_{y-axis[/tex] ⁰ R₉₀

e) An isometric transformation is a shape-preserving transformation in the plane or in space.

The isometric transformations are reflection, rotation and translation and combinations of them such as the glide, which is the combination of a translation and a reflection.

f) Reflection does not preserve orientation because this transformation changes vertices.

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The radius of convergence of the series is 1/8.

The radius of convergence, r, of the given series can be found using the ratio test.

Taking the limit of the ratio of the (n+1)th and nth term as n approaches infinity gives:

lim |(8(n+1) -1)/(n+1)| / |8n-1)/n| = lim |(8n +7)/(n+1)| = 8

Since the limit is finite and less than 1, the series converges absolutely.

Therefore, the radius of convergence, r, is given by the formula r = 1/lim sup(|an|^1/n) where an is the nth term of the series.

Substituting the given values and simplifying, we get r = 1/8.

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[tex]f(4c+1)=9(4c+1)+5=36c+9+5=36c+14[/tex]

you can choose to add the common factor which is 2(18c+7)

Let's denote the cost of making one vegetable sandwich as x. Then we know that the cost of a vegetable burger is x + 12. The cost of a vegetable burger is 14.75 rupee

From the problem statement, we know that the cost of making one sandwich is 2.75 rupees, so we can set up the equation:

x = 2.75

Then the cost of a vegetable burger is:

x + 12 = 2.75 + 12 = 14.75

In summary, the cost of a vegetable burger from the school canteen is 14.75 rupees. We can find this by adding the cost of making one sandwich (2.75 rupees) to the extra cost of 12 rupees for the burger.

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Let's denote the cost of making one vegetable sandwich as x. Then we know that the cost of a vegetable burger is x + 12. The cost of a vegetable burger is 14.75 rupee

From the problem statement, we know that the cost of making one sandwich is 2.75 rupees, so we can set up the equation:

x = 2.75

Then the cost of a vegetable burger is:

x + 12 = 2.75 + 12 = 14.75

In summary, the cost of a vegetable burger from the school canteen is 14.75 rupees. We can find this by adding the cost of making one sandwich (2.75 rupees) to the extra cost of 12 rupees for the burger.

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Answer:

  y ≥ 2x +1

Step-by-step explanation:

You want the inequality whose graph is shown.

The line has a rise of 2 units for each 1 to the right, so its slope is ...

  m = rise/run = 2/1 = 2

The line crosses the y-axis at y = 1, so the y-intercept is ...

  b = 1

The boundary line is ...

  y = mx + b

  y = 2x + 1

The shading is above the solid line, so the inequality symbol is ≥.

The inequality for the graph is ...

  y ≥ 2x +1

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The smallest integer k that satisfies the given condition is k = 1. In Big O notation, this can be expressed as √n = O([tex]n^1[/tex]) or simply √n = O(n).

The smallest integer k such that √n = O([tex]n^k[/tex]) can be determined by comparing the growth rates of the functions. In this case, we want to find the value of k that makes the function √n grow at most as fast as n^k.
The square root function, √n, is less complex than any positive integer power of n. In other words, as n becomes large, any positive integer power of n will grow faster than the square root of n. To satisfy the condition √n = O([tex]n^k[/tex]), we need to find the smallest integer value of k such that[tex]n^k[/tex] grows faster than √n.
Since k must be an integer, the smallest possible value is k = 1. This means we are comparing the growth rates of √n and n^1 (which is simply n). As n becomes large, n will indeed grow faster than √n.

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Changing the seed value during testing and training of models would not have a huge impact on the model accuracy.

RANDOM NUMBER SEED

Seed values are used during model building mainly to ensure reproducibility. Setting a random state value means that output value generated for randomly generated events would be the same.

Features which impact accuracy of models include the amount of training set , number of features used in training , level of cleanliness of the data fed into the model and other factors which could impact how the model learns.

Therefore , random number seed does not determine how the model learns or captures information. Hence, having little to no impact on the accuracy of the model.

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To find the area of the surface, we will use the formula. Therefore, the area of the surface is $\frac{4\pi}{3}(b^2-a^2)$.

$A=\iint\limits_{S},dS$

where $S$ is the surface of the sphere inside the cylinder.

Since the surface of the sphere and the cylinder are both symmetric about the $z$-axis, we can use cylindrical coordinates.

$x=r\cos\theta, y=r\sin\theta, z=z$

The sphere has the equation $x^2+y^2+z^2=b^2$, so substituting the cylindrical coordinates we get:

$r^2+z^2=b^2$

The cylinder has the equation $x^2+y^2=a^2$, so substituting the cylindrical coordinates we get:

$r^2=a^2$

The limits of integration for $r$ are from $0$ to $a$, and for $\theta$ are from $0$ to $2\pi$. The limits of integration for $z$ are from $-\sqrt{b^2-r^2}$ to $\sqrt{b^2-r^2}$.

$A=\int\limits_{0}^{2\pi}\int\limits_{0}^{a}\sqrt{1+(\frac{\partial z}{\partial r})^2+(\frac{\partial z}{\partial\theta})^2},r,dr,d\theta$

$\frac{\partial z}{\partial r}=\frac{-r}{\sqrt{b^2-r^2}}$, and $\frac{\partial z}{\partial\theta}=0$.

$A=\int\limits_{0}^{2\pi}\int\limits_{0}^{a}\sqrt{1+\frac{r^2}{b^2-r^2}},r,dr,d\theta$

Letting $u=\frac{r^2}{b^2-r^2}$, we have $\frac{du}{dr}=\frac{2b^2}{(b^2-r^2)^2}$, and so $du=\frac{2b^2}{(b^2-r^2)^2},r,dr$. Substituting $u$ and $du$ we get:

$A=\int\limits_{0}^{2\pi}\int\limits_{0}^{\frac{a^2}{b^2-a^2}}\sqrt{1+u},du,d\theta$

Using the substitution $v=1+u$, we get $dv=du$, and so:

$A=\int\limits_{0}^{2\pi}\int\limits_{1}^{1+\frac{a^2}{b^2-a^2}}\sqrt{v},dv,d\theta$

$A=\int\limits_{0}^{2\pi}\frac{2}{3}(1+\frac{a^2}{b^2-a^2})^{\frac{3}{2}},d\theta$

$A=\frac{4\pi}{3}(b^2-a^2)$

Therefore, the area of the surface is $\frac{4\pi}{3}(b^2-a^2)$.

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The statement is false. One-way ANOVA is a statistical test used to compare the means of three or more groups that are independent of each other. However, it does not assume that the populations have the same variance.

Instead, one of the assumptions of one-way ANOVA is that the populations being compared have equal variances, which means that the variation within each group is the same. This assumption is called homogeneity of variances or homoscedasticity.

If the populations do not have equal variances, it can lead to biased results and inaccurate conclusions. In such cases, a modified version of one-way ANOVA called Welch's ANOVA can be used, which does not assume equal variances among the groups.

To test for the homogeneity of variances assumption in one-way ANOVA, researchers can use statistical tests such as Levene's test or Bartlett's test. These tests assess whether the variances of the groups are significantly different from each other. If the results of these tests are significant, it indicates that the assumption of equal variances has been violated, and a modified version of ANOVA should be used instead.

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The linear regression equation is ŷ = 1.47x - 41.67

The projected test grade is 2.43

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

The grade (x) and test grade (y) scores

The linear regression equation can be calculated using a graphing tool, where we have the following summary:

The regression equation is

ŷ = bx + a

Where

b = SP/SSX = 463.5/316 = 1.46677

a = MY - bMX = 77.88 - (1.47*81.5) = -41.66693

So, we have

ŷ = 1.47x - 41.67

For the test grade 30, we have

ŷ = 1.47 * 30 - 41.67

Evaluate

ŷ = 2.43

Hence, the projected test grade is 2.43

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The probability of E is not given, but P(E U F) can be calculated using the formula P(E U F) = P(E) + P(F) - P(E ∩ F).

From the given information, P(E ∩ F) = P(F|E) * P(E) = 0.6 * P(E) and P(E|F) = P(E ∩ F) / P(F) = 0.012 / P(F). Using Bayes' theorem, P(F|E) = P(E|F) * P(F) / P(E) = 0.06 * P(F) / P(E), which can be simplified to P(F) = 0.1 * P(E). Substituting these values into the formula for P(E U F), we get P(E U F) = P(E) + 0.1 * P(E) - 0.012 = 1.1 * P(E) - 0.012. Therefore, we cannot determine if E and F are independent without knowing the probability of E.

Two events E and F are independent if and only if P(E ∩ F) = P(E) * P(F). In this case, we have P(E ∩ F) = 0.012, which is not equal to P(E) * P(F) = P(E) * 0.1 * P(E) = 0.1 * P(E)^2, since P(E) is not given. Therefore, we cannot conclude that E and F are independent.

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Therefore, the approximate value of the solution at t = 0.2 using the improved Euler's method with h = 0.1 is y ≈ 0.6994.

To approximate the solution of the differential equation y' = 1 - 2t^3y, with the initial condition y(0) = 0.5, at t = 0.2 using the improved Euler's method with h = 0.1, we can follow these steps:

Step 1: Initialize the values

Let t0 = 0 and y0 = 0.5 be the initial values.

Let h = 0.1 be the step size.

Let N = (t - t0) / h = (0.2 - 0) / 0.1 = 2 be the number of iterations.

Step 2: Iterate using the improved Euler's method

For i = 1 to N:

Calculate the slope at the current point using the equation:

k1 = 1 - 2t^3y

Calculate the value of y at the midpoint using the equation:

ymid = y0 + h * k1

Calculate the slope at the midpoint using the equation:

k2 = 1 - 2(t + h/2)^3 * ymid

Update the value of y using the equation:

y1 = y0 + h * k2

Update the value of t:

t = t0 + i * h

Update the value of y0 for the next iteration:

y0 = y1

Step 3: Calculate the approximate value of y at t = 0.2

After N iterations, the value of y at t = 0.2 will be y1.

Let's calculate the values using the above steps:

Iteration 1:

t = 0

y0 = 0.5

k1 = 1 - 2(0)^3 * 0.5 = 1

ymid = 0.5 + 0.1 * 1 = 0.6

k2 = 1 - 2(0.05)^3 * 0.6 = 0.99999375

y1 = 0.5 + 0.1 * 0.99999375 = 0.599999375

Iteration 2:

t = 0.1

y0 = 0.599999375

k1 = 1 - 2(0.1)^3 * 0.599999375 = 0.99800000624

ymid = 0.599999375 + 0.1 * 0.99800000624 = 0.6997994375

k2 = 1 - 2(0.15)^3 * 0.6997994375 = 0.99437500348

y1 = 0.599999375 + 0.1 * 0.99437500348 = 0.699437537823

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Answer:

pick the first one

Step-by-step explanation:

The following characteristics are true of the graph of all proportional relationships. The graph is linear. The line of the graph passes through the origin. The slope of the line is the constant of proportionality.

the only graph that goes through the origin (0,0) is the first one. it doesn't have an intercept.

so pick y=(2/3)x

Answer:

First choice:  y = 2/3x

Step-by-step explanation:

A linear equation is proportional if it is straight AND passes through the origin (0,0).  The only equation that meets this definition is the first one,

y = 2/3x

The probability of observing the sequence r, b, b, r, r in the given scenario is 1/120.

To find the probability of observing the sequence r, b, b, r, r in the given scenario, we can break it down step by step:

Probability of drawing the first ball as red (r):

Since the urn initially contains one red and one black ball, the probability of drawing a red ball is 1/2.

Probability of drawing the second ball as black (b) after the first ball was red:

After drawing the first red ball, it is returned to the urn along with another red ball, so the urn now contains two red balls and one black ball.

The probability of drawing a black ball is 1/3.

Probability of drawing the third ball as black (b) after the previous sequence was r, b:

After drawing the second black ball, it is returned to the urn along with another black ball, so the urn now contains two red balls and two black balls.

The probability of drawing a black ball is now 2/4 = 1/2.

Probability of drawing the fourth ball as red (r) after the previous sequence was r, b, b:

After drawing the third black ball, it is returned to the urn along with another black ball, so the urn still contains two red balls and two black balls.

The probability of drawing a red ball is 2/4 = 1/2.

Probability of drawing the fifth ball as red (r) after the previous sequence was r, b, b, r:

After drawing the fourth red ball, it is returned to the urn along with another red ball, so the urn now contains three red balls and two black balls.

The probability of drawing a red ball is 3/5.

To find the overall probability of observing the sequence r, b, b, r, r, we multiply the probabilities of each individual step:

[tex]P(r, b, b, r, r) = (1/2) \times (1/3) \times (1/2) \times (1/2) \times (3/5)[/tex]

= 1/120.

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All of the criteria's are fulfilled the survey is valid.

We have the information from the question:

The proportion of college football players who have had at least one concussion is estimated to be 34% in the united states.

Then, 34% = 0.34

The sample size of the data is = 100

p: the ‘proportion’ of ‘college’

The required conditions for testing the hypothesis of population proportion are,

(i) The population is larger than the sample

(ii) np > 10

=> 100 × 0.34

      =34

(iii) n(1-p) > 10

100 × (1 - 0.34)

=> 100 × 0.66

      =66

iv)The ‘sample’ is drawn randomly from the population.

Since all of the above criteria's are fulfilled the survey is valid.

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Answer:

[tex]\frac{3}{26}[/tex] of pulling a queen or a black 3.

Step-by-step explanation:

Since there's 52 cards in an average deck and there's 4 queens, divide 52 with 4 and you'll get 13, so that means it's a [tex]\frac{1}{13}[/tex] chance of getting a queen.

Also because there's 4 "3"s and 50% of those cards are black (clubs and spades), divide 52 with 2 and you'll get a [tex]\frac{1}{26}[/tex] chance of getting a black 3.

Add both quotients to a common denominator of 52 and you will end up with 6/52. Then simplify the sum.

You should end up with [tex]\frac{3}{26}[/tex].

Answer:

To begin, we set each component to zero:

x + 3 = 0 x^2 - 4 = 0

When we solve for x in the first equation, we get:

x = -3

We may factor the second equation further using the difference of squares formula which is:

(x + 2)(x - 2) = 0

Then, in each factor, we solve for x:

x + 2 = 0 or x - 2 = 0

x = -2 or x = 2

As a result, the function's zeroes are x = -3, x = -2, and x = 2.

Answer:

-3, -2, and 2.

Step-by-step explanation:

Solving for x in the first equation gives:

x+3 = 0

x = -3

Solving for x in the second equation gives:

x^2-4 = 0

(x+2)(x-2) = 0

x+2 = 0 or x-2 = 0

x = -2 or x = 2

Therefore, the zeroes of the function F(x) are -3, -2, and 2.

The probability of exactly 1 seed not growing is 31.36%. we can use the binomial probability formula, which is:

P(X=k) = (n choose k) * p^k * (1-p)^(n-k). P(X=k) is the probability of getting k successes in n trials n is the number of trials

k is the number of successes p is the probability of success in each trial.

To calculate the probability of exactly 1 failure, we can use the binomial probability formula, which is:

P(X=k) = (n choose k) * p^k * (1-p)^(n-k)

P(X=k) is the probability of getting k successes in n trials

n is the number of trials

k is the number of successes

p is the probability of success in each trial

(n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items

In this case, n=6, k=1, p=0.65, and (n choose k) = 6. Plugging these values into the formula, we get:

P(X=1) = (6 choose 1) * 0.65^1 * (1-0.65)^(6-1) = 6 * 0.65 * 0.35^5 = 0.3136

Therefore, the probability of exactly 1 seed not growing is 31.36%.

In this case, the probability of getting exactly 1 failure in 6 trials, where the probability of success is 0.65. We use the binomial coefficient to count the number of ways to choose 1 item (the failure) from a set of 6 items (the seeds). We then multiply this by the probability of getting 1 failure and 5 successes, which is given by the formula p^k * (1-p)^(n-k), where p is the probability of success and k and n-k represent the number of successes and failures, respectively. Finally, we calculate the probability of exactly 1 failure by multiplying the binomial coefficient and the probability of getting 1 failure and 5 successes.

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Answer:A

Step-by-step explanation:

using the raiload racks

The equations for each object's velocity, in meters/second, in terms of its mass in kilograms are:

[tex]V_A=\sqrt{\frac{200}{m} }\\\\V_B=\sqrt{\frac{50}{m} }[/tex]

A graph of the two functions is shown below.

In Mathematics, the kinetic energy of an object can be calculated by using the following equation (formula):

K.E = 1/2 × mv²

Where:

By making velocity (v) the subject of formula, we have:

[tex]V= \sqrt{\frac{2K.E}{m} }[/tex]

In this context, the equations for each object's velocity, in terms of its mass can be written as follows;

Velocity of object A = [tex]V_A= \sqrt{\frac{2(100)}{m} }[/tex]

Velocity of object A = [tex]V_A= \sqrt{\frac{200}{m} }[/tex]

Velocity of object B = [tex]V_B= \sqrt{\frac{2(25)}{m} }[/tex]

Velocity of object B = [tex]V_B= \sqrt{\frac{50}{m} }[/tex]

When mass (m) = 2 kg, the velocity of object A can be calculated as follows;

[tex]V_A=\sqrt{\frac{200}{m} } \\\\V_A=\sqrt{\frac{200}{2} }\\\\V_A=10 \;m/s[/tex]

When mass (m) = 2 kg, the velocity of object B can be calculated as follows;

[tex]V_B=\sqrt{\frac{50}{m} } \\\\V_B=\sqrt{\frac{50}{2} }\\\\V_B=5 \;m/s[/tex]

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Complete Question:

The kinetic energy, E₁, in kilograms meters squared per second squared (kg.m²/sec) of an object can be modeled with the equation E, = 1/2mv².

where m is the object's mass in kilograms and v is the object's velocity in meters per second.

A physics student is investigating 2 moving objects:

Object A's kinetic energy is 100 kg m²/sec².

Object B's kinetic energy is 25 kg m²/sec²

Write equations for each object's velocity, in meters/second, in terms of its mass in kilograms. Then graph the two functions on the same coordinate grid. Provide evidence to support your answer.

To find the margin of error, we need to first determine the formula for it. The margin of error (ME) is calculated by multiplying the critical value of the confidence level (in this case 90%) with the standard error (SE) of the sample mean.

The critical value for a 90% confidence interval can be found using a t-distribution table with n-1 degrees of freedom (where n is the sample size). For a sample size of 30, the degrees of freedom would be 29. Using the table, the critical value for a 90% confidence interval is approximately 1.697.

Next, we need to calculate the standard error. The formula for the standard error of the sample mean is the standard deviation of the population divided by the square root of the sample size. However, we don't know the standard deviation of the population, so we will use the sample standard deviation as an estimate.

Assuming that the sample is representative of the population, we can assume that the sample standard deviation is an unbiased estimate of the population standard deviation. Therefore, we can use the formula:

SE = s / sqrt(n)

where s is the sample standard deviation and n is the sample size.

Since we are not given the sample standard deviation, we cannot calculate the standard error. However, we can use the range of the confidence interval to estimate it. The range of the confidence interval is equal to the margin of error multiplied by 2. Therefore:

ME = (51.3 - 48.7) / 2 = 1.3

Using the formula for the margin of error, we can solve for the standard error:

ME = t*SE

1.3 = 1.697*SE

SE = 0.767

Therefore, the margin of error is approximately 1.3 inches.

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The speed of the river's current is -27/2 or -13.

let's use d to represent the distance between their starting point and the point 6 miles downstream and r to represent the speed of the river's current. since they were able to paddle downstream, they must have been going faster than the speed of the river's current. let's call their downstream speed s1. similarly, their upstream speed would have been slower than the speed of the river's current, so let's call their upstream speed s2. using the formula distance = rate x time, we can write two equations based on the given information:equation 1: d = (s1 + r) x (40/60)     (since they paddled downstream for 40 minutes)equation 2: d = (s2 - r) x 3           (since they paddled upstream for 3 hours)

we can solve for s1 and s2 by adding and subtracting equation 1 and equation 2:d = (s1 + r) x (40/60)d = (s2 - r) x 32d = (s1 + r) x (40/60) + (s2 - r) x 3simplifying this equation, we get:

2d = (s1 + r) x (2/3) + (s2 - r) x 32d = (2s1 + 2r + 3s2 - 3r) / 36d = 2s1 + 2r + 3s2 - 3r6d = 2s1 + 3s2 - rnow we can use equation 1 to substitute s1 + r with d x (3/8):

d = (s1 + r) x (40/60)d = (s1 + r) x (2/3)s1 + r = d x (3/4)substituting this expression into the previous equation, we get:6d = 2(d x (3/4)) + 3s2 - r6d = (3d/2) + 3s2 - r

9d/2 = 3s2 - rr = 3s2 - (9d/2)now we need to find s2, which we can do by using equation 2:d = (s2 - r) x 3s2 = (d/3) + r

substituting r with the previous expression, we get:s2 = (d/3) + 3s2 - (9d/2)s2/3 = -3d/2s2 = -9d/2finally, we can substitute this value of s2 into the expression for r:

r = 3s2 - (9d/2)r = -27d/2 5 miles per hour. however, since this answer is negative, it does not make physical sense.

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