A rectangle has a length of 10 inches and a width of 3 inches whose sides are changing. The length is increasing by 6 in/sec and the width is shrinking at 3 in/sec. What is the rate of change of the area?

The following parts can  be answered by the concept from Standard deviation.

a. We need a sample size of at least 121 for each group.

b. We need a sample size of at least 78 for each group.

c.  We need a sample size of at least 97.48 for each group.

To find the sample size needed to estimate (P1-P2) for each of the given situations, we can use the following formula:

n = (Zα/2)² × (p1 × q1 + p2 × q2) / (P1 - P2)²

where:
- Zα/2 is the critical value of the standard normal distribution at the desired confidence level
- p1 and p2 are the estimated proportions in the two populations
- q1 and q2 are the complements of p1 and p2, respectively (i.e., q1 = 1 - p1 and q2 = 1 - p2)
- (P1 - P2) is the desired margin of error

a. For a margin of error equal to 0.11 with 99% confidence, assuming p1 ~ 0.6 and p2 ~ 0.4, we have:

Zα/2 = 2.576 (from standard normal distribution table)
p1 = 0.6, q1 = 0.4
p2 = 0.4, q2 = 0.6
(P1 - P2) = 0.11

Plugging in the values, we get:

n = (2.576)² × (0.6 × 0.4 + 0.4 × 0.6) / (0.11)²
n ≈ 120.34

Therefore, we need a sample size of at least 121 for each group.

b. For a 90% confidence interval of width 0.88, assuming no prior information is available to obtain approximate values of p1 and p2, we have:

Zα/2 = 1.645 (from standard normal distribution table)
(P1 - P2) = 0.88
Since we have no information about p1 and p2, we can assume them to be 0.5 each (which maximizes the sample size and ensures a conservative estimate).

Plugging in the values, we get:

n = (1.645)² × (0.5 × 0.5 + 0.5 × 0.5) / (0.88)²
n ≈ 77.58

Therefore, we need a sample size of at least 78 for each group.

c. For a margin of error equal to 0.08 with 90% confidence, assuming p1 = 0.19 and p2 = 0.3, we have:

Zα/2 = 1.645 (from standard normal distribution table)
q1 = 0.81
q2 = 0.7
(P1 - P2) = 0.08

Plugging in the values, we get:

n = (1.645)² × (0.19 × 0.81 + 0.3 × 0.7) / (0.08)²
n ≈ 97.48

Therefore, we need a sample size of at least 98 for group 1. For group 2, we can use the same sample size as group 1, or we can adjust it based on the expected difference between p1 and p2 (which is not given in this case).

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The following parts can  be answered by the concept from Standard deviation.

a. We need a sample size of at least 121 for each group.

b. We need a sample size of at least 78 for each group.

c.  We need a sample size of at least 97.48 for each group.

To find the sample size needed to estimate (P1-P2) for each of the given situations, we can use the following formula:

n = (Zα/2)² × (p1 × q1 + p2 × q2) / (P1 - P2)²

where:
- Zα/2 is the critical value of the standard normal distribution at the desired confidence level
- p1 and p2 are the estimated proportions in the two populations
- q1 and q2 are the complements of p1 and p2, respectively (i.e., q1 = 1 - p1 and q2 = 1 - p2)
- (P1 - P2) is the desired margin of error

a. For a margin of error equal to 0.11 with 99% confidence, assuming p1 ~ 0.6 and p2 ~ 0.4, we have:

Zα/2 = 2.576 (from standard normal distribution table)
p1 = 0.6, q1 = 0.4
p2 = 0.4, q2 = 0.6
(P1 - P2) = 0.11

Plugging in the values, we get:

n = (2.576)² × (0.6 × 0.4 + 0.4 × 0.6) / (0.11)²
n ≈ 120.34

Therefore, we need a sample size of at least 121 for each group.

b. For a 90% confidence interval of width 0.88, assuming no prior information is available to obtain approximate values of p1 and p2, we have:

Zα/2 = 1.645 (from standard normal distribution table)
(P1 - P2) = 0.88
Since we have no information about p1 and p2, we can assume them to be 0.5 each (which maximizes the sample size and ensures a conservative estimate).

Plugging in the values, we get:

n = (1.645)² × (0.5 × 0.5 + 0.5 × 0.5) / (0.88)²
n ≈ 77.58

Therefore, we need a sample size of at least 78 for each group.

c. For a margin of error equal to 0.08 with 90% confidence, assuming p1 = 0.19 and p2 = 0.3, we have:

Zα/2 = 1.645 (from standard normal distribution table)
q1 = 0.81
q2 = 0.7
(P1 - P2) = 0.08

Plugging in the values, we get:

n = (1.645)² × (0.19 × 0.81 + 0.3 × 0.7) / (0.08)²
n ≈ 97.48

Therefore, we need a sample size of at least 98 for group 1. For group 2, we can use the same sample size as group 1, or we can adjust it based on the expected difference between p1 and p2 (which is not given in this case).

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If an estimated regression line has a y-intercept of 10 and a slope of 4, then when x = 2 the actual value of y is 18, the actual value of y remains unknown.

When working with an estimated regression line, we typically use the equation y = b0 + b1x, where y is the dependent variable (the value we want to predict), x is the independent variable, b0 is the y-intercept, and b1 is the slope of the line.

In this case, the estimated regression line has a y-intercept (b0) of 10 and a slope (b1) of 4. So, the equation of the line is y = 10 + 4x.

Now, you want to know the actual value of y when x = 2. To find the estimated value of y, plug x = 2 into the equation:

y = 10 + 4(2) = 10 + 8 = 18.

However, it's important to note that the estimated regression line is only an approximation of the relationship between x and y. It does not provide the exact value of y for a given x; instead, it provides a prediction based on the observed data used to generate the line. In reality, there may be other factors influencing the value of y that are not accounted for by the regression line.

So, while the estimated value of y when x = 2 is 18, the actual value of y remains unknown. It could be close to the estimated value or significantly different, depending on the degree of variation in the data and any additional factors that may affect the relationship between x and y.

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The size of the reflex angle POR is 302 degrees.

Since the angle PQR is given as 29 degrees and it lies on the circumference of the circle, we know that it is an inscribed angle that intercepts the arc PR. The measure of an inscribed angle is half the measure of the intercepted arc. Therefore, we can find the measure of the arc PR as:

Arc PR = 2 × Angle PQR = 2 × 29 = 58 degrees

Since angle POR is a reflex angle that contains the inscribed angle PQR and the arc PR, we can find its measure by subtracting the measure of angle PQR from 360 degrees:

Angle POR = 360 - Arc PR = 360 - 58 = 302 degrees

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The Confidence interval for the population proportion p is approximately 0.4832

To determine the lower bound of a two-sided 95% confidence interval for the population proportion p, we can use the formula for the confidence interval of a proportion.

The formula for the confidence interval of a proportion is given by:

CI = p ± zsqrt((p(1-p))/n)

where:

CI = confidence interval

p = sample proportion

z = z-score corresponding to the desired confidence level

n = sample size

Given:

Sample proportion (p) = 37/80 = 0.4625 (since 37 out of 80 bicycle wheels were found to have critical flaws)

Sample size (n) = 80

Desired confidence level = 95%

We need to find the z-score corresponding to a 95% confidence level. For a two-sided confidence interval, we divide the desired confidence level by 2 and find the z-score corresponding to that area in the standard normal distribution table.

For a 95% confidence level, the area in each tail is (1 - 0.95)/2 = 0.025. Using a standard normal distribution table or a z-score calculator, we can find that the z-score corresponding to an area of 0.025 is approximately -1.96.

Now we can plug in the values into the formula and solve for the lower bound of the confidence interval:

CI = 0.4625 ± (-1.96)sqrt((0.4625(1-0.4625))/80)

Calculating the expression inside the square root first:

(0.4625*(1-0.4625)) = 0.2497215625

Taking the square root of that:

sqrt(0.2497215625) ≈ 0.4997215107

Substituting back into the formula:

CI = 0.4625 ± (-1.96)*0.4997215107

Now we can calculate the lower bound of the confidence interval:

Lower bound = 0.4625 - (-1.96)*0.4997215107 ≈ 0.4625 + 0.979347415 ≈ 1.4418 (rounded to four decimal places)

Therefore, the lower bound of the two-sided 95% confidence interval for the population proportion p is approximately 0.4418 (rounded to four decimal places).

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n^2

all the numbers on the right are squares of the numbers on the left

squares means the number times the same number

Answer:

Number 2, [tex]n^{2}[/tex]

Step-by-step explanation:

The table shows at the top of the screen has a very specific pattern, when comparing side length and area.

When the side length is 4 the area is 16

When the side length is 5 the area is 25

What is happening?

They are being squared(Multipled by itself).

See here:

4*4 = 16

5*5 = 25

Understand how the table is working?

The table is a side to area comparision of a polygon.

The question asks to find the area of a similar polygon, if a side length is n.

Because we are squaring the side length, the answer is:

[tex]n^{2}[/tex]

The expected value of X is then calculated as E(X) = ∫x f(x) dx from 0 to 1, which simplifies to E(X) = k∫x⁵ dx from 0 to 1. Evaluating this integral gives us the expected value of X, which is equal to 5/6.

The expected value of the random variable X with a probability density function (pdf) of f(x) = kx⁴ is calculated as E(X) = ∫x f(x) dx from negative infinity to positive infinity.

Integrating f(x) from negative infinity to positive infinity gives us the normalizing constant k, which is equal to 1/∫x⁴ dx from 0 to 1. Simplifying this gives us k = 5.

The expected value of X is then calculated as E(X) = ∫x f(x) dx from 0 to 1, which simplifies to E(X) = k∫x⁵ dx from 0 to 1. Evaluating this integral gives us E(X) = k/6, which is equal to 5/6. Therefore, the expected value of X with f(x) = kx⁴ pdf is 5/6.

In summary, the expected value of a random variable X with a probability density function (pdf) of f(x) = kx⁴ is calculated by integrating x f(x) from negative infinity to positive infinity. Integrating f(x) from negative infinity to positive infinity gives us the normalizing constant k, which is equal to 1/∫x⁴ dx from 0 to 1.

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The given statement "For a second-order homogeneous linear ordinary differential equation (ODE), an initial value problem (IVP) consists of an equation and two initial conditions" is True because  A second-order homogeneous linear ODE is an equation of the form ay''(t) + by'(t) + cy(t) = 0, where y(t) is the dependent variable, t is the independent variable, and a, b, and c are constants.

The equation is homogeneous because the right-hand side is zero, and it is linear because y(t), y'(t), and y''(t) are not multiplied or divided by each other or their higher powers. An IVP for this type of equation requires two initial conditions because the second-order ODE has two linearly independent solutions.

These initial conditions are typically given in the form y(t0) = y0 and y'(t0) = y1, where t0 is the initial time, and y0 and y1 are the initial values of y(t) and y'(t), respectively.

The two initial conditions are necessary to determine a unique solution to the second-order ODE. Without them, there would be an infinite number of possible solutions. By providing the initial conditions, you establish constraints on the solutions, which allow for a unique solution that satisfies both the ODE and the initial conditions.

In summary, an IVP for a second-order homogeneous linear ODE consists of an equation and two initial conditions, ensuring a unique solution to the problem.

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Answer:

6x(x+6)(x+10)

Step-by-step explanation:

6x^(3)+96x^(2)+360x

x6(x^2+16x+60)

6x(x+6((x+10)

When finding a confidence interval for a population mean based on a sample of size 8, the assumption made is that the sampled population is approximately normal.

When finding a confidence interval for a population mean based on a sample of size 8, the assumption made is that the sampled population is approximately normal. This assumption is crucial because it ensures that the sampling distribution of the sample mean is normal or nearly normal, allowing for accurate confidence interval calculations.

This assumption allows us to use the central limit theorem, which states that the distribution of sample means will approach a normal distribution as the sample size increases. This in turn allows us to use a t-distribution to calculate the confidence interval.

Option A is incorrect because the sampling distribution of z is used when the population standard deviation is known, which is not the case in this scenario. Option B is also incorrect because assumptions are made in statistical inference. Option C is incorrect because it assumes that the population standard deviation is known, which is not always the case.

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The area of the rear tire of the tractor is A = 201.1 feet²

Given data ,

The area of a circle is given by the formula A = πr², where r is the radius of the circle.

Given that the radius of the tractor tire is 8 feet, we can substitute this value into the formula to calculate the area:

A = π(8²)

Using the value of π as approximately 3.14159265359

A ≈ 3.14159265359 x (8²)

A = 3.14159265359 x 64

A ≈ 201.061929829746

Rounding to the nearest tenth, we get:

A ≈ 201.1 feet²

Hence , the area of the tractor tire is approximately 201.1 feet²

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The change of coordinates matrix P from the basis B to the basis C is given by P = [[-23/42, -15/42], [-46/42, 30/42]], which simplifies to P = [[-23/42, -5/14], [-23/21, 5/7]].

To find the change of coordinates matrix P from basis B to basis C, follow these steps:

1. Write the basis vectors of B and C as column vectors: B = [[42], [-15]] and C = [[-23], [1-2]].

2. Find the inverse of the matrix formed by basis B, B_inv = (1/determinant(B)) * adjugate(B). The determinant of B is -630, so B_inv = (1/-630) * [[-15, 15], [-42, 42]] = [[15/630, -15/630], [42/630, -42/630]] = [[1/42, -1/42], [2/30, -2/30]].

3. Multiply the matrix B_inv with matrix C to obtain the change of coordinates matrix P: P = B_inv * C = [[1/42, -1/42], [2/30, -2/30]] * [[-23], [1-2]] = [[-23/42, -15/42], [-46/42, 30/42]] = [[-23/42, -5/14], [-23/21, 5/7]].

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It is generally recommended to keep the load factor below 0.75 for hash tables using either linear probing or chaining.

No, neither linear probing nor chaining can function properly with a load factor greater than 1.

When the load factor exceeds 1, it means that the number of items in the hash table exceeds the number of available buckets, and collisions become unavoidable.

In linear probing, this results in an endless loop of searching for an empty bucket, making it impossible to insert new items or retrieve existing ones.

In chaining, a high load factor can cause the chains to become very long, slowing down retrieval operations significantly.

In extreme cases, the chains can become so long that the hash table degenerates into a linked list, rendering the hash table useless.

Therefore, it is generally recommended to keep the load factor below 0.75 for hash tables using either linear probing or chaining.

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Answer:

7306.1

Step-by-step explanation:

The value of the car is $7306.10 after 8 years.

Given

A new car is purchased for 16600 dollars.

The value of the car depreciates at 9.75% per year.

What is depreciation?

Depreciation denotes an accounting method to decrease the cost of an asset.

To get the depreciation of a partial year, you need to calculate the depreciation a full year first.

The formula to calculate depreciation is given by;

V= P( 1-r )^t

Where V represents the depreciation r is the rate of interest and t is the time.

Hence, the value of the car is $7306.10 after 8 years.

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The fourth graph represents the functions f(x)=-3ˣ-2

We can plug in the y intercept to find which graph has the correct one.

x = 0 is y intercept

Thus function f(0)=-3⁰-2

f(0)=-1-2

f(0)=-3

At this point we known the y intercept is -3 so both graph in the left is considerable.

Notice that the base is the negative, thus the graph would goes down. Therefore the bottom right would be correct.

Hence, the fourth graph represents the functions f(x)=-3ˣ-2

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The area of the figure in this problem is given as follows:

140 yd².

The figure in the context of this problem is a composite figure, hence the area is the sum of the areas of all the parts that compose the figure.

The figure in this problem is composed as follows:

The area of each part of the figure is given as follows:

Hence the total area of the figure is given as follows:

100 + 40 = 140 yd².

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The solution to the nonhomogeneous differential equation y′′+9y=cos(3x)+sin(3x) with initial conditions y(0)=3 and y′(0)=1 is y(x) = c1*cos(3x) + c2*sin(3x) + (1/6)*x*sin(3x) - (1/18)*cos(3x).


Step 1: Find the complementary function, y_h, which is the general solution to the associated homogeneous equation y'' + 9y = 0. The characteristic equation is r^2 + 9 = 0, so r = ±3i. Hence, y_h = c1*cos(3x) + c2*sin(3x).

Step 2: Find a particular solution, y_p, to the nonhomogeneous equation. Assume y_p = A*cos(3x) + B*sin(3x) + C*x*cos(3x) + D*x*sin(3x). Plug this into the nonhomogeneous equation and simplify to determine A, B, C, and D. We get A=-1/18, B=0, C=0, D=1/6.

Step 3: Combine the complementary function and particular solution: y(x) = y_h + y_p = c1*cos(3x) + c2*sin(3x) - (1/18)*cos(3x) + (1/6)*x*sin(3x).

Step 4: Apply initial conditions to find c1 and c2. y(0) = 3 => c1 = 3 + 1/18, y'(0) = 1 => c2 = 1/6. Thus, y(x) = (3+1/18)*cos(3x) + (1/6)*sin(3x) + (1/6)*x*sin(3x) - (1/18)*cos(3x).

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The equivalent expression of the logarithmic expression (logₓy)² is log²ₓy

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

(logₓy)²

The above expression is pronounced

log y to the base of x all squared

When the expression is expanded, we have the following

(logₓy)² = (logₓy) * (logₓy)

Evaluating the expression, we have

(logₓy)² = log²ₓy

Hence, the equivalent expression of the expression (logₓy)² is log²ₓy

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Answer:

C

Step-by-step explanation:

7 + 45/5 = 16

Correct. The assumption that the sample size should be at least 25 is correct. This is because, for a sample to be representative of the population, it should have enough observations to provide a reasonable estimate of the population parameters.

A sample size of at least 25 is generally considered the minimum requirement for statistical analysis. The statement about the remaining assumption is correct. In order to make valid inferences from a random sample, it is important to have a large enough sample size (n). A common rule of thumb is that the sample size should be at least 25. This helps to ensure that the sample is representative of the population and increases the accuracy of the results.

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Answer:

∴ The other integer is -2099.

Step-by-step explanation:
Let the unknown number be x,

599+x=(-1500)

x=(-1500)-599

x=(-2099)

The magnitude and the direction of the vectors  v=⟨−12,5⟩ in degrees for the condition 0 ≤ θ < 360 is equal to 13 and -22.62 degrees respectively.

Let us consider two vectors named v₁ and v₂.

Here, in degrees

0 ≤ θ < 360

v=⟨−12,5⟩

This implies that

The value of the vector 'v₁' = -12

The value of the vector 'v₂' = 5

Magnitude of the vectors v₁ and v₂ is equals to

=√ ( v₁ )² + ( v₂)²

Substitute the values of the  vectors v₁ and v₂ we get,

⇒Magnitude of the vectors v₁ and v₂ = √ (-12 )² + ( 5)²

⇒Magnitude of the vectors v₁ and v₂ = √144 + 25

⇒Magnitude of the vectors v₁ and v₂ = √169

⇒Magnitude of the vectors v₁ and v₂ = 13

Direction of the vectors for the condition 0 ≤ θ < 360 defined by

θ = tan⁻¹ ( v₂ / v₁ )

⇒ θ = tan⁻¹ ( 5 / -12 )

⇒ θ = -22.62 degrees.

Therefore, the magnitude and the direction of the vectors is equal to 13 and -22.62 degrees respectively.

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a. To find the standard error of the sampling distribution of sample means:

Standard deviation = sqrt(Variance of the population)

Since the population standard deviation is not given, we assume it is 1.

Standard error = (Standard deviation) / sqrt(n)

        = (1) / sqrt(100)

        = 0.01  (rounded to 2 decimal places)

b.

Standard error = 0.01  (from part a)

z = (4.7 - mean) / 0.01

        = (4.7 - 5) / 0.01

        = -0.3  (rounded to 2 decimal places)

Chance that sample mean is between 4.7 and 5.9 hours

        = P(z > -0.3) + P(z < 0.3)

        = 0.762 + 0.761

        = 0.7524  (rounded to 4 decimal places)

c.

Standard error = 0.01  (from part a)

z = (5.1 - mean) / 0.01

        = 0.1  (rounded to 2 decimal places)

Chance that sample mean is between 5.1 and 5.5 hours

        = P(z > 0.1) + P(z < -0.1)

        = 0.4583 + 0.4603

        = 0.4593  (rounded to 4 decimal places)

d.

Standard error = 0.01  (from part a)

z = (7.60 - mean) / 0.01

        = 3  (rounded to 2 decimal places)

Chance that sample mean is greater than 7.60 hours

        = P(z > 3)

        = 0  (rounded to 4 decimal places)

This would be very unlikely.

The correct answer to the question is "You think the 1968 minimum wage was at most $10.86 when, in fact, it was not."

Explanation: -

In statistical hypothesis testing, a Type I error is the rejection of a null hypothesis when it is actually true.

In this scenario, the null hypothesis is that the 1968 minimum wage is p≤$10.86. If a researcher thinks that the 1968 minimum wage was at most $10.86, but in reality, it was not, this would be a Type I error. In other words, the researcher rejected the null hypothesis (that the minimum wage was $10.86 or less) when it was actually true.

To determine the probability of making a Type I error, we use the significance level, denoted by α. The significance level is the probability of rejecting the null hypothesis when it is actually true. If we set α=0.05, this means that there is a 5% chance of making a Type I error. So, if we reject the null hypothesis that the 1968 minimum wage is $10.86 or less, when in fact, it is true, we are making a Type I error with a probability of 0.05 or 5%.

Therefore, the correct answer to the question is "You think the 1968 minimum wage was at most $10.86 when, in fact, it was not."

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The common ratio 'r' is not constant, meaning that the series is not geometric.

Each term in a geometric series is created by multiplying the previous term by a fixed constant known as the common ratio.

To determine if the geometric series (4, -7, 49, -343, 16) is convergent or divergent, we need to find the common ratio 'r' of the series.

r = (next term) / (current term)

r = (-7) / 4 = -1.75

r = 49 / (-7) = -7

r = (-343) / 49 = -7

r = 16 / (-343) = -0.0466...

We can see that the common ratio 'r' is not constant, meaning that the series is not geometric, and therefore we cannot determine if it is convergent or divergent.

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The value of x in the intersection of chords is 15.

option A.

The value of x is calculated by applying the following formula as shown below;

Based on intersecting chord theorem, the arc angle formed at the circumference due to  intersection of two chords, is equal to half the tangent angle.

∠RFE = ¹/₂ x 104⁰

∠ RFE = 52

The sum of ∠GFE  = 90 (line GE is the diameter)

∠GFE = ∠GFR + ∠RFE

90 = (x + 23) + 52

90 = x + 75

x = 90 - 75

x = 15

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X in r2 whose coordinate vector relative to the basis b is [1/5 2/15].

To find x⃗ in r2 whose coordinate vector relative to the basis b is [2 -4], we first need to express the basis vectors as a matrix.

The matrix for the basis b is:
[ -4 12
 -5  0 ]

To find x⃗, we can use the formula:
x⃗ = [x⃗ ]b * [B]^-1
where [B]^-1 is the inverse of the matrix for the basis b.

To find the inverse of the matrix for the basis b, we can use the formula:
[B]^-1 = (1/60) * [0 12
                    5 -4 ]

Plugging in the values, we get:
x⃗ = [2 -4] * (1/60) * [0 12
                              5 -4 ]
  = (1/60) * [(-8)+(20) (24)+(-16)]
  = (1/60) * [12 8]
  = [1/5 2/15]

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