Find \sin(\angle BAC + 30^{\circ})sin(∠BAC+30


∘


)sine, left parenthesis, angle, B, A, C, plus, 30, degrees, right parenthesis

Gabriela will use 1620 square inches of wood to build the open box.

The amount of wood Gabriela will use depends on the surface area of the box, which is the sum of the areas of its six faces. Since the box is open on top, it will have five faces: four sides and a bottom.

The area of the bottom is the area of a rectangle with length 18 in. and width 15 in., which is:

Area of bottom = length x width = 18 in. x 15 in. = 270 in²

The area of each side is the product of the height and the length of the corresponding base, which is:

Area of each side = height x length = 15 in. x 18 in. = 270 in²

So the total surface area of the box is:

Total surface area = 2 x (Area of bottom) + 4 x (Area of each side)

= 2 x 270 in² + 4 x 270 in²

= 1620 in²

Therefore, Gabriela will use 1620 square inches of wood to build the open box.

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None of the bills have an experimental probability of 3, as all probabilities are between 0 and 1.

Based on the table, the experimental probability for each bill being drawn from the bag next can be calculated by dividing the frequency of each bill by the total number of draws (100). Using this formula, we can calculate the experimental probabilities for each bill:

1. For the $1 bill: Experimental probability = [tex]\frac{(Frequency of  $1 bill)}{Total draws} = \frac{8}{100} = 0.28[/tex]
2. For the $10 bill: Experimental probability =[tex]\frac{(Frequency of  $10 bill)}{Total draws} = \frac{56}{100} = 0.56[/tex]
3. For the $20 bill: Experimental probability =[tex]\frac{(Frequency of  $20 bill)}{Total draws} = \frac{2}{100} = 0.02[/tex]

None of the bills have an experimental probability of 3, as all probabilities are between 0 and 1.

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Answer:

  36 units

Step-by-step explanation:

You want to know the perimeter of the right triangle with leg lengths 9 units and 12 units.

The perimeter of the triangle is the sum of the lengths of its sides. We can count squares to find the lengths of the horizontal and vertical legs. The length of the hypotenuse can be found using the Pythagorean theorem:

  c² = a² +b²

  c² = 9² +12² = 81 +144 = 225

  c = √225 = 15 . . . . length of the hypotenuse

Then the perimeter is ...

  P = a +b +c = 9 +12 +15 = 36 . . . units

The perimeter of the triangle is 36 units.

__

Additional comment

The leg lengths have the ratio 9:12 = 3:4, telling you this is a 3:4:5 right triangle. This means you know the side lengths are 3:4:5 = 9:12:15, and their sum is 9+12+15 = 36.

It is handy to memorize a few of the Pythagorean triples that often show up in algebra, trig, and geometry problems: {3, 4, 5}, {5, 12, 13}, {7, 24, 25}, {8, 15, 17}.

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The combination of side lengths that would not form a triangle is C.XY = 11 mm, YZ = 14 mm, XZ = 21 mm.

We shall use the triangle inequality theorem to determine if a set of side lengths can form a triangle.

The triangle inequality theorem says that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

We shall calculate each of the options:

For option A:

XY + YZ = 7 mm + 14 mm = 21 mm which is < XZ = 25 mm.

Therefore, option A does form a triangle.

For option B:

XY + YZ = 11 mm + 18 mm = 29 mm, which is > XZ = 21 mm.

YZ + XZ = 18 mm + 21 mm = 39 mm, which is > XY = 11 mm.

XY + XZ = 11 mm + 21 mm = 32 mm, which is > YZ = 18 mm.

Therefore, option B does form a triangle.

For option C:

XY + YZ = 11 mm + 14 mm = 25 mm, and is > XZ = 21 mm.

Therefore, option C does not form a triangle.

For option D:

XY + YZ = 7 mm + 14 mm = 21 mm, which is > XZ = 17 mm.

YZ + XZ = 14 mm + 17 mm = 31 mm, which is > XY = 7 mm.

XY + XZ = 7 mm + 17 mm = 24 mm, which is > YZ = 14 mm.

Therefore, option D does form a triangle.

Therefore, the combination of side lengths that would not form a triangle is XY = 11 mm, YZ = 14 mm, XZ = 21 mm.

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Answer:

no it is not possible

Step-by-step explanation:

For the function "h(x) = |2x| - 8", the domain is (-∞, ∞) and the range is  [-8, ∞).

The function h(x) = |2x| - 8 is defined for all real numbers x, so the domain of h(x) is the set of all real-numbers, or (-∞, ∞).

To find the range of the function, we determine set of all possible output values of function. Since the function involves the absolute value of 2x, the output can never be less than -8.

When "2x" is positive, |2x| = 2x. When 2x is negative, |2x| = -2x. This means that the function h(x) will have two branches depending on whether 2x is positive or negative.

⇒ When 2x is positive, h(x) = |2x| - 8 = 2x - 8. This branch of the function will have all non-negative values.

⇒ When 2x is negative, h(x) = |2x| - 8 = -2x - 8. This branch of the function will have all non-positive values.

Combining the two , we get the range of the function h(x) as [-8, ∞).

Therefore, the domain of h(x) is (-∞, ∞) and the range of h(x) is [-8, ∞).

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The average number of customers in the shop is 7.5

The average number of customers in the shop can be calculated using the M/M/2 queuing model. In this model, we assume that the arrivals follow a Poisson distribution, and the service times follow an exponential distribution.

The subscript "2" in M/M/2 refers to the fact that there are two servers or service channels available.

Using Little's Law, the average number of customers in a stable system is equal to the product of the arrival rate and the average time spent in the system.

Thus, to calculate the average number of customers in the shop, we need to find the average time spent in the system.

The average time spent in the system can be calculated as the sum of the average time spent waiting in the queue and the average time spent being served. Using the M/M/2 queuing model,

the average time spent waiting in the queue can be calculated as [tex](λ^2)/(2μ(μ-λ))[/tex], where λ is the arrival rate and μ is the service rate. In this case, λ=3 and μ=1/2 since there is one barber who can serve one customer at a time.

Thus, the average time spent waiting in the queue is [tex](3^2)/(21/2(1/2-3))[/tex] = 9/4 hours. The average time spent being served is the mean service time, which is 1/4 hour. Therefore, the average time spent in the system is 9/4 + 1/4 = 5/2 hours.

Finally, using Little's Law, the average number of customers in the shop is λ times the average time spent in the system, which is 3*(5/2) = 15/2 or 7.5 customers.

However, since the shop can only accommodate at most two customers at a time, the actual number of customers in the shop would be either one or two.

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Sea "l" la medida del largo del terreno y "a" la medida del ancho del terreno.

De acuerdo con el problema, el ancho es la mitad del largo, es decir, a = l/2.

El perímetro de un rectángulo se calcula sumando las longitudes de sus cuatro lados, por lo que en este caso:

Perímetro = 2l + 2a = 2l + 2(l/2) = 3l

Sabemos que el perímetro es de 105 metros, entonces:

3l = 105

l = 105/3 = 35

Por lo tanto, el largo del terreno es 35 metros. Y, como el ancho es la mitad del largo, entonces:

a = l/2 = 35/2 = 17.5

Por lo tanto, el ancho del terreno es de 17.5 metros.

En resumen, las medidas del terreno son 35 metros de largo y 17.5 metros de ancho.

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The appropriate measures of center and variation are 13 and 4.

Measures of Center:

The appropriate measure of center for this data set is the median since there is no clear outlier present in the data. Hence, the value of median here is 13

Measures of Variation:

The appropriate measure of variation for this dataset is the range, which is the difference between the largest and smallest value in the dataset, Hence the value of range is 4

Since the data is small and there is no clear outlier present, the median is the appropriate measure of center. The range, which is the difference between the largest and smallest value in the dataset, is the appropriate measure of variation

Hence, the appropriate measures of center and variation are 13 and 4.

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The composite function fg(x) is a linear function​ by the proof shown below

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

f(x) = 2x - 1

g(x) = -x + 3

The above functions are linear functions

This means that the function fg(x) will also be a linear function​

To prove this, we have

f(g(x)) = 2(g(x)) - 1

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

f(g(x)) = 2(-x + 3) - 1

So, we have

f(g(x)) = -2x - 7

Hence, the function is a linear function

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Yes, any ramp with an angle of 4.8 degrees will have a slope ratio of 1:12.

The slope ratio is the ratio of the vertical rise to the horizontal run of the ramp, and it is equivalent to the tangent of the angle of inclination of the ramp.

The tangent of 4.8 degrees is approximately 0.0084, which means that for every 1 unit of vertical rise, there is 0.0084 units of horizontal run. To convert this to a ratio, we can multiply both sides by 100 to get:

1 unit of rise : 100 x 0.0084 = 0.84 units of run

Simplifying this ratio by dividing both sides by 0.84, we get:

1 unit of rise : 1.19 units of run

which is equivalent to a slope ratio of 1:12 (since 12 = 1/0.084). Therefore, any ramp with an angle of 4.8 degrees will have a slope ratio of 1:12.

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The surface area of the similar cone is 415.8 in²

A cone is a shape formed by using a set of line segments or the lines which connects a common point, called the apex or vertex.

The surface area of a cone is expressed as;

SA = πr( r+l) where r is the radius and l is the slant height.

The slant height of the original cone =

l= √h²+r²

l = √12²+3²

l = √144+9

l = √153

l = 12.4 in

SA= 3π( 3+12.4)

SA = 3 × 15.4

SA = 46.2 in²

The surface area of similar cone with radius 9 inches is calculated by;

(3/9)² = 46.2/x

= 9/81 = 46.2/x

x = 46.2 × 81/9

x = 415.8in²

Therefore the surface area of the similar cone is 415.8 in³

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I think it might be C

The expression that represents the fee for signing up the day of the activity is x + 0.20x. This means that fee for signing up the day of the activity is original fee (x) plus an additional 20% of original fee (0.20x).

For example, if the community center charges $50 for signing up before the day of the activity, then the fee for signing up the day of the activity would be:

$50 + ($50 x 0.20) = $50 + $10 = $60

So, individuals who sign up the day of the activity are charged a higher fee than those who sign up before the day of the activity, because they are charged the original fee plus an extra 20% of the original fee. This is often used as an incentive for individuals to sign up early and to help the community center plan and prepare for the activity accordingly.

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It will take the rock 1.5 seconds  to hit the ground

Since the situation is modeled by the equation h = -16t² + 16t + 12

The rock will hit the ground when height is zero i.e. h = 0. Thus, we have:

-16t² + 16t + 12 = 0 and we can then solve for the time (t) .

-16t² + 16t + 12 = 0  (divide through by -16)

t² - t - 3/4 = 0

Using factorization method:

(t + 1/2) (t - 3/2) = 0

t = -1/2 or 3/2

Since t cannot be negative. Thus, t = 3/2 = 1.5.

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We can use the formula for the area of a sector to find the area of sector $LMN$

The area of a sector with central angle $\theta$ in a circle with radius $r$ is given by:

$A = \frac{\theta}{360^\circ} \pi r^2$

In this case, we know that $m\angle LMN = 66^\circ$ and $LM = 19$ units, so the radius of circle M is half of the diagonal of the rectangle formed by $LM$ and $MN$. Using the Pythagorean theorem, we can find the length of $MN$:

$MN^2 = LM^2 + LN^2 = LM^2 + LM^2 = 2LM^2$

$MN = \sqrt{2} LM = \sqrt{2} \cdot 19$

So the radius of circle M is $r = \frac{1}{2}MN = \frac{1}{2}\sqrt{2} \cdot 19$

Now we can use the formula for the area of a sector to find the area of sector $LMN$:

$A = \frac{m\angle LMN}{360^\circ} \pi r^2 = \frac{66^\circ}{360^\circ} \pi \left(\frac{1}{2}\sqrt{2} \cdot 19\right)^2 \approx \boxed{90.89}$ square units (rounded to the nearest hundredth).

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The minimum number of people that need to be persuaded is two. When there are 0 contributor, 1 contributor, 2 or more contributor this is a the Nash equilibria.

a. Let's first calculate the benefit to each citizen when there are n contributors. According to the problem, the benefit is n^2 dollars. So when there are 0 contributors, the benefit to each citizen is 0 dollars. When there is 1 contributor, the benefit to each citizen is 1 dollar. When there are 2 contributors, the benefit to each citizen is 4 dollars. And so on, up to 10,000 dollars per citizen when all 100 citizens contribute.

Now let's think about the incentives of each citizen to contribute. If no one contributes, everyone gets 0 dollars of benefit. If one person contributes, that person gets 1 dollar of benefit, and everyone else gets 0 dollars. So each person has an incentive to free-ride, hoping that someone else will contribute.

But if two people contribute, each person gets 4 dollars of benefit, which is more than the 1 dollar cost of contributing. So once there are at least two contributors, it becomes rational for everyone else to contribute as well.

Therefore, the minimum number of people that need to be persuaded is two. Once two people contribute, it becomes rational for everyone else to contribute as well.

b. Let's consider the Nash equilibria of the game where each citizen is deciding whether to contribute. A Nash equilibrium is a situation where no one has an incentive to change their strategy, given the strategies of all the other players.

In this case, each citizen has two strategies: contribute or free-ride. Let's consider the case where n citizens are contributing. If everyone else is contributing, then it is rational to contribute as well, since the benefit of contributing is greater than the cost.

If everyone else is free-riding, then it is rational to free-ride as well, since the cost of contributing is greater than the benefit. However, if some people are contributing and some people are free-riding, then it may be rational to contribute, since the benefit of contributing may outweigh the cost, depending on the number of contributors.

Therefore, there are multiple Nash equilibria in this game, depending on the number of contributors. When there are 0 contributors, everyone is free-riding and this is a Nash equilibrium. When there is 1 contributor, that person is contributing and everyone else is free-riding, and this is a Nash equilibrium. When there are 2 or more contributors, everyone is contributing, and this is a Nash equilibrium.

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The probability that a person either runs outside or joins a gym is 0.60 or 60%.

To find the probability that a person either runs outside or joins a gym, you can use the formula:

P(A or B) = P(A) + P(B) - P(A and B)

where A is the event of running outside, B is the event of joining a gym, and "and" represents the intersection of both events.

Given:
P(running outside) = 0.45
P(joining a gym) = 0.40
P(both running outside and joining a gym) = 0.25

Plug these values into the formula:

P(either runs or joins a gym) = 0.45 + 0.40 - 0.25

Calculate the result:

P(either runs or joins a gym) = 0.60

Therefore, the 60% of person either runs outside or joins a gym

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Answer:

Volume = 288695.6 in³

Step-by-step explanation:

Volume of a sphere is given by

v=4/3πr^3

Where r is the radius of the sphere

From the question

radius = 41 in

Substitute the value into the above formula

We have

v=4/3 x 41^3π

=275684/3 π

= 288695.6097

We have the final answer as

Volume = 288695.6 in³ to the nearest tenth

Volume = 288695.6 in³ to the nearest tenth

Hope this helps you

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The choice which best describes the measure of center of these data is (d) Rome data center is "best-described" by median and  data center of "New-York" is "best-described" by mean.

In the table provided below, we see that the "IQR" for Rome is relatively large compared to the IQR for New York, which suggests that that there may be some skewness in the distribution of the Rome data. So, the median would be a better-measure of center than the mean.

On the other hand, the IQR for "New-York" is relatively small, which indicates that the data is more symmetric and the mean would be a better measure of center.

Therefore, the correct answer is (d).

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The given question is incomplete, the complete question is

The table shows data from a survey about the number of times families eat at restaurants during a week. The families are either from Rome, Italy, or New York, New York:

                    High   Low    Q1     Q3    IQR    Median    Mean    σ

Rome              18       1         3       7        4         6.5          6.4    4.3

New York       14       1        4.5    8.5     4          5.5          6.1     3.2

Which of the choices below best describes how to measure the center of these data?

(a) Both centers are best described by the mean.

(b) Both centers are best described by the median.

(c) The Rome data center is best described by the mean. The New York data center is best described by the median.

(d) The Rome data center is best described by the median. The New York data center is best described by the mean.

To evaluate the line integral JF.dr, where F = (-2 sin x, 2 cos y, 6zx) and C is the path given by r(t) = (3t^3, -3t^2, -3t) for 0 <= t <= 1, we first need to parameterize F and r in terms of t.

For F, we have:

F = (-2 sin x, 2 cos y, 6zx) = (-2 sin (3t^3), 2 cos (-3t^2), 6(3t^3)(-3t)) = (-2 sin (3t^3), 2 cos (3t^2), -54t^4)

For r, we already have the parameterization:

r(t) = (3t^3, -3t^2, -3t)

Now we can use the formula for the line integral:

JF.dr = ∫(F dot dr)

= ∫(-2 sin (3t^3) dx + 2 cos (3t^2) dy - 54t^4 dz)

= ∫(-18t^2 cos (3t^2) + 18t^2 cos (3t^2) - 54t^4) dt

= ∫(-54t^4) dt

= -9t^5 + C

Evaluating this expression for t = 1 and t = 0, we get:

JF.dr = (-9(1)^5 + C) - (-9(0)^5 + C)

= -9 + 9

= 0

Therefore, the line integral JF.dr evaluated along the path given by r(t) = (3t^3, -3t^2, -3t) for 0 <= t <= 1 is equal to 0.

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The solution to the system of linear equations is x = 1, y = -3, and z = 8, which is option B: X=-1, y=-3, z=2.

To solve this system of linear equations, we can use Gaussian elimination, which involves adding and subtracting equations to eliminate variables. Here are the steps:

x - 3y - 2z = 6

2x - 4y - 3z = 8

-3x + 6y + 8z = -5

Step 1: Add twice the first equation to the second equation to eliminate x:

x - 3y - 2z = 6

4y + z = 20

-3x + 6y + 8z = -5

Step 2: Add three times the first equation to the third equation to eliminate x:

x - 3y - 2z = 6

4y + z = 20

9y + 2z = 13

Step 3: Solve for z in the second equation:

4y + z = 20

z = 20 - 4y

Step 4: Substitute z into the third equation and solve for y:

9y + 2z = 13

9y + 2(20 - 4y) = 13

y = -3

Step 5: Substitute y into the second equation and solve for z:

4y + z = 20

4(-3) + z = 20

z = 8

Step 6: Substitute y and z into the first equation and solve for x:

x - 3y - 2z = 6

x - 3(-3) - 2(8) = 6

x = 1

Therefore, the solution to the system of linear equations is x = 1, y = -3, and z = 8, which is option B: X=-1, y=-3, z=2.

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The text format of the question in the picture:

15. The solution the system of linear equation of

x-3y-2z = 6

2x-4y-3z = 8

(-3x+6y+8z = -5 is

A) X=-1,y=-3, z=-2 B) X=-1,y=-3, z=2 C) X=1,y=-3, z=2 D) X = 1, y = 3, z=-2

For a random sample of 155 observations results in 62 successes.

a) A 90% confidence interval for the population proportion of successes is equals to the (0.335 , 0.465).

b) A 90% confidence interval for the population proportion of failure is equals to the (0.535 , 0.665).

We have a random sample of 155 observations results in 62 successes. So,

Observed value, x = 62

Sample size,n = 155

Population Proportion, p = x/n

= 62/155

= 0.4

a) We have to determine 90% confidence interval for the population proportion of successes. Using the distribution table, for 90% confidence interval, z-score value is equals to 1.6. Consider Confidence interval formula with proportion, CI [tex]= p ± z×\sqrt\frac{p(1-p)}{n}[/tex]

substitute all known values in above formula, [tex]= 0.4 ± 1.64\sqrt\frac{0.4(1- 0.4)}{155}[/tex]

= 0.4 ± 0.0645

= (0.4 - 0.0645 , 0.4 + 0.0645)

= (0.335 , 0.465)

b) Now, we have to determine a 90% confidence interval for the population proportion of failures.

Now consider, here failure observed values, x = 155 - 62

= 93

proportion, p = x/n

= 93/155 = 0.6

Consider the confidence interval formula, CI [tex]= p ± z×\sqrt\frac{p(1-p)}{n}[/tex]

substitute values, [tex]= 0.6 ±1.64×\sqrt\frac{0.6(1-0.6)}{155}[/tex]

= 0.6 ± 0.0645

= (0.6 - 0.0645 , 0.6 + 0.0645)

= (0.535 , 0.665)

Hence, required value is (0.535 , 0.665).

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The fish has located a horizontal distance of 7 meters away from the cliff.

How to find the distance between the bird and the fish?

To find the distance between the bird and the fish, we need to find the horizontal distance traveled by the bird during the dive. We can do this by finding the x-coordinate of the vertex of the parabolic curve, which represents the highest point of the dive.

The vertex of the parabolic curve of the given function f(x) = (x-7)^2 - 1 is at the point (7, -1). This means that the highest point of the bird's dive is reached at 7 seconds, and the bird is at a height of -1 meters at this point.

To find the distance traveled by the bird during the dive, we need to find the horizontal distance between the bird's starting point (the cliff) and the highest point of the dive (the vertex). The distance is given by the horizontal coordinate of the vertex, which is 7 seconds.

Therefore, the fish has located a horizontal distance of 7 meters away from the cliff, assuming that the bird started the dive from the edge of the cliff.

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Corbin's account will have a higher balance than Victoria's account at the end of 3 years" is true.

To calculate the balance at the end of 3 years, we can use the simple interest formula for Victoria's account and the compound interest formula for Corbin's account.

For Victoria's account:

Simple interest = P * r * t

= 2000 * 0.05 * 3

= $300

Balance after 3 years = P + Simple interest

= 2000 + 300

= $2300

For Corbin's account:

Balance after 3 years = [tex]P * (1 + r)^t[/tex]

= 1800 * (1 + 0.09)³

= $2401.40

Therefore, the statement "Corbin's account will have a higher balance than Victoria's account at the end of 3 years" is true.

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The coordinate matrix of the linear transformation with respect to the second ordered basis is found by multiplying the change of basis matrix by the coordinate matrix of the linear transformation with respect to the first ordered basis is [tex]\left[\begin{array}{cc}0&2/5\\1&-1/5\end{array}\right][/tex]

Let V be a vector space with two ordered bases B and B', and let T be a linear transformation from V to V. Suppose that the change of basis matrix from B to B' is P, and the coordinate matrix of T with respect to B is A.

To find the coordinate matrix of T with respect to B', we can use the formula

A' = P⁻¹AP

where A' is the coordinate matrix of T with respect to B'.

To use this formula, we need to find the inverse of P. If P is invertible, then we have

P⁻¹ = 1/det(P) * adj(P)

where det(P) is the determinant of P and adj(P) is the adjugate of P.

Assuming that P is invertible, we can compute its inverse as follows

det(P) = 1*(-2) - 2*2 = -5

adj(P) =[tex]\left[\begin{array}{cc}-2&2\\-2&1\end{array}\right][/tex]

So, P⁻¹ = (-1/5)*[tex]\left[\begin{array}{cc}-2&2\\-2&1\end{array}\right][/tex] = [tex]\left[\begin{array}{cc}2/5&-2/5\\2/5&-1/5\end{array}\right][/tex]

Now, we can use the formula to find the coordinate matrix of T with respect to B'

A' = P⁻¹AP =  *[tex]\left[\begin{array}{cc}-2&1\\-1&0\end{array}\right][/tex]*[tex]\left[\begin{array}{cc}-1&2\\2&1\end{array}\right][/tex]= [tex]\left[\begin{array}{cc}0&2/5\\1&-1/5\end{array}\right][/tex]

Therefore, the coordinate matrix of T with respect to B' is

[tex]\left[\begin{array}{cc}0&2/5\\1&-1/5\end{array}\right][/tex]

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--The given question is incomplete, the complete question is given

"  Using change of base matrices to find coordinate matrices of linear transformations Let B and C be two ordered bases of R2, and consider a linear transformation T: R2 + R2. Suppose that the change of base matrix Ic, B is given by [0 -2 3 3] and the coordinate matrix Tc,c of T with respect to C is given by [ -1 -1 2 -1] Use this to determine coordinate matrix TB,B of T with respect to B. TB,B ? "--

Answer:

(-1/5)x + 8 = -x + 4

(4/5)x + 8 = 4

(4/5)x = -4

x = -5, so y = 9

C. There is one unique solution (-5, 9).

The crop sizes for first 4 years without using fertilisers are:

The size of the potato crop for each of the first four years is computed as follows:

For Year 1:

= 15 tonnes * 1

= 15 tonnes

For Year 2:

= 15 tonnes x 0.8

= 12 tonnes

For Year 3:

= 12 tonnes x 0.8

= 9.6 tonnes

For Year 4:

= 9.6 tonnes x 0.8

= 7.68 tonnes

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We can expect  4 giant swallowtails to visit Ellie's garden.

In total, there were 96 butterfly visits (32 monarchs + 56 gulf fritillaries + 8 giant swallowtails).

Now, we can calculate the probability of a giant swallowtail visit:
Probability

= (number of giant swallowtails) / (total butterfly visits)

= 8/96

= 1/12

Given that 48 butterflies visit her garden, we can expect the number of giant swallowtails to be:
Expected giant swallowtails

= (total butterflies) × (probability of giant swallowtails)

= 48 × (1/12)

= 4

So, we can expect approximately 4 giant swallowtails to visit Ellie's garden out of the 48 butterflies.

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