16 ft
Find the area.
20 ft
15 ft A = [?] ft²
Round to the nearest
hundredth.
12ift
10 ft
Remember: A = πr² Use 3.14 for .

Given sin∠X = cos∠Y and m∠X = 72°, we can find the measure of angle Y to be 18° since angles X and Y are complementary.

The problem states that sin∠X = cos∠Y and m∠X = 72°, and we are asked to find the measure of angle Y.

The first thing to notice is that sin∠X = cos∠Y means that the sine of angle X is equal to the cosine of angle Y. By the definition of sine and cosine, we know that:

sin∠X = opposite/hypotenuse

cos∠Y = adjacent/hypotenuse

where "opposite" and "adjacent" are the lengths of the sides of a right triangle that correspond to angles X and Y, respectively, and "hypotenuse" is the length of the hypotenuse of the triangle.

Since sin∠X = cos∠Y, we can set the two expressions equal to each other:

sin∠X = cos∠Y

opposite/hypotenuse = adjacent/hypotenuse

opposite = adjacent

This tells us that the lengths of the opposite and adjacent sides of the right triangle are equal. Since these sides are opposite and adjacent to angles X and Y, respectively, this means that angles X and Y are complementary angles (i.e., the sum of their measures is 90°).

We know that angle X has a measure of 72°, so we can use the fact that angles X and Y are complementary to find the measure of angle Y:

m∠Y = 90° - m∠X

m∠Y = 90° - 72°

m∠Y = 18°

Therefore, the measure of angle Y is 18°.

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After computing the composition of f(g(x)) and g(f(x)), it is clear that f(x) and g(x) are not inverse functions of each other.

To determine whether f(x) and g(x) are inverse functions of each other, we need to check whether the composition of the two functions f(g(x)) and g(f(x)) result in the identity function, which is equal to x.

First, let's find f(g(x)):

f(g(x)) = f(5x - 5/4) (substituting g(x) into f(x))

f(g(x)) = 4/5(5x - 5/4) + 1 (substituting the expression for f(x))

f(g(x)) = 4x - 1 + 1

f(g(x)) = 4x

Now let's find g(f(x)):

g(f(x)) = g(4/5x + 1) (substituting f(x) into g(x))

g(f(x)) = 5(4/5x + 1) - 5/4 (substituting the expression for g(x))

g(f(x)) = 4x + 5 - 5/4

g(f(x)) = 4x + 20/4 - 5/4

g(f(x)) = 4x + 15/4

Since f(g(x)) = 4x and g(f(x)) = 4x + 15/4, we can see that the two compositions are not equal to x, which means that f(x) and g(x) are not inverse functions of each other.

Therefore, we can conclude that f(x) and g(x) are not inverse functions of each other.

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The volume of the cone with radius as 9 inches and height as 11 inches is 933 cubic inches.

The formula for the volume of a cone is the product of the following multiplication, given as V = π×r²×h/3.

In this formula, π is the pie or 22/7, r is the radius, and h is the height.

The radius of the base of cone, r = 9 inches

The height of the cone, h = 11 inches

π = 22/7

The volume of the cone = πr²h/3 cubic units.

= 22/7 x 9 x 9 x 11/3

= 22/7 x 81 x 3.667

= 933 cubic inches

= 933 in³

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The value of x is 15 when y =8

If x varies indirectly as y, then we can write:

x = k/y

where k is the constant of variation. To find the value of k, we can use the given information that x = 5 when y = 24:

5 = k/24

Multiplying both sides by 24, we get:

k = 120

Now we can use this value of k to find x when y = 8:

x = 120/8 = 15

Therefore, the answer is A. 15.

A ratio that depicts the association between the independent variable (x) and the dependent variable is known as a constant of variation (k) (y). In the event that both of those variables have known values, it can be calculated by dividing y by x.

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Answer:

Step-by-step explanation:

We can solve the equation 4x+2 = 12 for x using the change of base formula by isolating x on one side of the equation:

4x + 2 = 12

Subtracting 2 from both sides:

4x = 10

Dividing both sides by 4:

x = 2.5

We do not need to use the change of base formula to solve this equation as it is a simple linear equation that can be solved by basic algebraic manipulations. Therefore, none of the given options are correct.

Part a: Model for m∠A : cos A = 200/d.

Part b: If d = 300 m,  m∠A = 48.19°.

Part c: If d = 550 m, m∠A =  68.67° .

The trigonometric functions sine, cosine, tangent, and its inverses make up these trigonometric ratios.

Given an angle measurement, so every function represents a distinct ratio in either a right triangle. Trigonometric ratios are the ratios of a right triangle's sides to one of the triangle's non-right angles. Remember that a right triangle has one right angle that is always 90 degrees in length.

In the given query:

Base  = 200 m.

Hypotenuse = d m.

Part a: Model for m∠A :

cos A = Base/ Hypotenuse

cos A = 200/d.

Thus, Model for m∠A : cos A = 200/d.

Part b: If d = 300 m,  m∠A = ?

cos A = 200/d.

cos A = 200/300

cos A = 2/3

m∠A =  48.19°

Part c: If d = 550 m, m∠A = ?

cos A = 200/d.

cos A = 200/550

cos A = 4/11

m∠A = 68.67°

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Answer:

5

Step-by-step explanation:17+24=41

41/8=5.125

So The least you can get for the banquet table is 5.

Step-by-step explanation:

For RIGHT triangles , remember  S-O-H-C-A-H-T-O-A

sin 14° = opposite leg / hypotenuse

         sin 14° = x / 10

          10* sin 14° = x

               x = 2.42 units

sin 37° = 10 / a

a = 10 / sin 37°

a = 16.62 units

According to the question, it takes Mina 80 minutes to commute to work.

Two equations are considered to be comparable when their roots and solutions line up. To create an equivalent equation, the identical quantity, symbol, or expression must always be added to or removed from both of the equation's two sides. We can also create a similar equation simply multiplying or dividing either sides of the an equation by a nonnegative value.

Let's denote the time it takes Mina to commute to work by "m" (in minutes).

The issue states that Katie's train trip requires 10 minutes or more half as much time as Mina's drive. Instead, we might write:

Katie's commute time = (1/2) * Mina's commute time + 10

We also know that it takes Katie 30 minutes to get to work, so we can write:

Katie's commute time + 30 minutes = total time to get to work

The result of putting the very first equation into to the second equation is:

[(1/2) * Mina's commute time + 10] + 30 minutes = total time to get to work

Simplifying the equation, we get:

(1/2) * Mina's commute time + 40 minutes = total time to get to work

Now we can set this equation equal to "m" to solve for Mina's commute time:

(1/2) * m + 40 = m

Subtracting (1/2) * m from both sides, we get:

40 = (1/2) * m

Multiplying both sides by 2, we get:

m = 80

Therefore, it takes Mina 80 minutes to commute to work.

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Check the picture below.

so the path of the population P(t) is parabolic, more or less like the one in the picture, so it reaches its maximum at the vertex and at "t" time of the x-coordinate of the vertex.

[tex]\textit{vertex of a vertical parabola, using coefficients} \\\\ P(t)=\stackrel{\stackrel{a}{\downarrow }}{-1709}t^2\stackrel{\stackrel{b}{\downarrow }}{+80000}t\stackrel{\stackrel{c}{\downarrow }}{+10000} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)[/tex]

[tex]\left(-\cfrac{ 80000}{2(-1709)}~~~~ ,~~~~ 10000-\cfrac{ (80000)^2}{4(-1709)}\right) \implies \left( - \cfrac{ 80000 }{ -3418 }~~,~~10000 - \cfrac{ 6400000000 }{ -6836 } \right) \\\\\\ \left( \cfrac{ -40000 }{ -1709 } ~~~~ ,~~~~ 10000 + \cfrac{ 1600000000 }{ 1709 } \right) ~~ \approx ~~ (\stackrel{ hours }{\text{\LARGE 23}}~~,~~946220)[/tex]

Two airplanes are flying in the air at the same height. airplane a is flying east at 453 mi/h and airplane b is flying north at 508 mi/h. if they are both heading to the same airport, located 7 miles east of airplane a and 8 miles north of airplane b, at the rate at which the distance between the two airplanes is changing is approximately 473 mi/h.

What is the distance between the airplanes?

We may use the Pythagorean theorem to find the distance between the two airplanes. Let’s use A to represent the position of Airplane A and B to represent the position of Airplane B.

Let d be the distance between the airplanes. Then, using the Pythagorean Theorem, we have:

d² = (8 miles)² + (7 miles)² d² = 64 + 49d² = 113d = sqrt(113) miles

Since Airplane A is flying straight to the airport, its rate of approach to the airport is its speed, 453 mi/h.

Since Airplane B is flying straight to the airport, its rate of approach to the airport is its speed, 508 mi/h.

Let d be the distance between the airplanes at some point in time t. We need to find the rate at which the distance is changing, or the derivative of d with respect to t. We may use the Pythagorean theorem to find the distance between the two airplanes.

Let A represent the position of Airplane A and B represent the position of Airplane B.Let d be the distance between the airplanes. Then, using the Pythagorean Theorem, we have:

d² = (8 miles)² + (7 miles)² d² = 64 + 49d² = 113d = sqrt(113) miles

At some time t, let A(t) represent the position of Airplane A, and let B(t) represent the position of Airplane B. We have that:

A(t) = 453t B(t) = 508t

Therefore, the distance between the airplanes is given by:

d(t)² = (453t)² + (508t)²d(t)² = 205,609t² + 258,064t²d(t)² = 463,673t²

We take the derivative of both sides with respect to t, noting that d²/dt² = 2dd/dt:

2d(t)d'(t) = 927,346t

Then, dividing both sides by 2d(t), we have:

[tex]d'(t) = 927,346t/(2d(t))d'(t) = 927,346t/(2sqrt(113)) miles/h[/tex]

Using t = 1 hour (since we are asked for the rate at which the distance between the airplanes is changing), we have:

[tex]d'(1) = 927,346/(2sqrt(113))d'(1) ≈ 473 miles/h[/tex]

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Answer:

37.5%

Step-by-step explanation:

Based on the given conditions, formulate: 30/80

Reduce the fraction: 3/8

Rewrite a fraction as a decimal: 0.375

Multiply a number to both the numerator and the denominator:

0.375 * 100/100

Write as a single fraction:

0.375 * 100 / 100

Calculate the product or quotient:

37.5/100

Rewrite a fraction with denominator equals 100 to a percentage:

37.5%

Answer:

37.5%

According to the question the area of the triangle with sides of 10, 9, and 6 is 45.

Area is a term used to describe the size of a two-dimensional space. It is a measure of how much surface is contained within a two-dimensional boundary. It is usually expressed in units of square meters, square kilometers, square feet, or square miles. Area can also be used to describe the size of a three-dimensional space, such as for a volume. In this case, area is expressed in units of cubic meters, cubic feet, or cubic miles. Area is an important concept in mathematics and is used to calculate the area of shapes, such as triangles and circles, as well as to solve problems related to probability, motion, and other topics. Area is also used in physics to calculate the amount of force that an object exerts on another object.

The area of a triangle can be calculated using the formula A = 1/2 * b * h, where b is the base of the triangle and h is the height of the triangle.
Using this formula, the area of a triangle with a base of 10 and a height of 9 would be A = 1/2 * 10 * 9 = 45.
Therefore, the area of the triangle with sides of 10, 9, and 6 is 45.

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The area of the triangle with sides of lengths 10, 9, and 6 is approximately 37 square units.

What is the Pythagorean theorem?

In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

To find the area of a triangle, we can use the formula:

Area = 1/2 * base * height

In this case, we need to identify the base and height of the triangle.

The longest side of the triangle is 10, which is opposite to the largest angle, so we can take that as the base. Let's call it b.

To find the height, we need to draw an altitude from the vertex opposite the base. Let's call the height h.

Now, we can use the Pythagorean theorem to find the height:

h^2 = 10^2 - (9^2 + 6^2)/2^2

h^2 = 100 - 45.25

h^2 = 54.75

h = sqrt(54.75)

h = 7.4 (rounded to one decimal place)

Now that we have the base and height, we can use the formula to find the area:

Area = 1/2 * base * height

Area = 1/2 * 10 * 7.4

Area = 37 square units (rounded to the nearest whole number)

Therefore, the area of the triangle with sides of length 10, 9, and 6 is approximately 37 square units.

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Answer:

Given :

By selling an article for rupees 144, a man loses 17

of his outlay. It is sold for rupees 189.

To do :

We have to find the gain or loss percent.

Solution :

Let the cost price of article be Rs. x

.

This implies,

SP =CP −Loss

=x−x7

=6x7

Therefore,

6x7=144

x=144×76

x=Rs. 168

If the article is sold for Rs. 189, then,

SP =Rs. 189

Profit =

SP − CP

=189−168

=Rs. 21

Gain %=GainCP×100%

=21168×100%

=12.5%

The gain percent is 12.5%

.

Step-by-step explanation:

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a) The plot that represents a dot plot of the data is plot (B).

b) The answer is (C)

c) The answer is (B)

A normal distribution is a continuous probability distribution that has a symmetric bell-shaped curve, with the mean, median, and mode all being equal.

(a) The plot that represents a dot plot of the data is plot (B).

(b) The configuration of the points does not suggest that the volumes are from a population with a normal distribution. The answer is (C) - The dot plot does not approximate a "bell" form, hence the population does not seem to have a normal distribution.

(c) There are no outliers in the data. The answer is (B) - No, there does not appear to be any outliers.

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The answers are:

1). B. The plot that depicts a data dot plot is plot (B).

2). C. Because the dotplot does not match a "bell" shape, the population does not appear to have a normal distribution.

3). B. No, there does not appear to be any outlie

A normal distribution is a kind of continuous distribution of probability in which the majority of data points cluster in the centre of the range, while the remainder taper off symmetrically towards either extreme. The mean of the distribution is also known as the centre of the range.

Because of its flared form, a normal distribution resembles a bell curve graphically. The exact shape can vary depending on the population's value distribution. The population is the total number of data elements in the distribution.

a). The plot depicts a data dot plot and is called plot. (B).

b). Because the dot plot does not resemble a "bell" form.(C).

c). The data does not contain any anomalies. (B).

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The Complete question is,

a). question a is attached below.

b). Does the configuration of the points appear to suggest that the volumes are from a population with a normal distribution?

A. Yes, the population appears to have a normal distribution because the dotplot resembles a "bell shape

B. No, the population does not appear to have a normal distribution because the frequencies of the volume decrease from left to right.

C. No, the population does not appear to have a normal distribution because the dotplot does not resemble a "bell" shape

D. Yes, the population appears to have a normal distribution because the frequencies of the volume increase from left to right.

c). Are there any outliers?

A. Yes, the volumes of 0 oz and 200 oz appear to be outliers because they are far away from the other temperatures

B. No, there does not appear to be any outliers °

C. Yes, the volume of 50 oz appears to be an outlier because it is far away from the other volumes

D. Yes, the volume of 70 oz appears to be an outlier because many sodas had this as their volume

General equation of line is [tex]y=mx+n[/tex] where m is slope and n is point on y-axis. So just use points in question to determine what m and n must be. Let me show you.

For A(0,1), put this point in [tex]y=mx+n[/tex] then you have [tex]1=m.0+n[/tex] Hence [tex]n=1[/tex]

Now use second one that is B(1,5), then you get [tex]5=m.1+1[/tex] since [tex]n=1[/tex]. Finally you get [tex]m=4[/tex] that is slope.

Therefore, D is the correct answer.

Answer:

Step-by-step explanation:

I’m not good at explaining but basically if you do 20n - 15n it would be 5n

Add £48.28 to £142 so you get £190.28

AnswerAnswerAnswerAnswer:

190.28

Step-by-step explanation:

£142 + 34% = £142 x 1.34 = 190.28

For the given function f(x)=  [tex]-8(2)^{3x}[/tex] + 3 which contains variable x , whose value on substituting as zero is found to be (calculated without using calculator)

What is variable?

Variable is a term used in algebra or algebraic expressions and equations to represent the unknown values or whose value is not fixed. variables and constants are combined to form algebraic expressions or equations. The difference between expression and an equation is that expressions do not contain 'equal to' sign and equations shows balance between left hand side and the right side using 'equal to' sign.

Here the function is f(x)=  [tex]-8(2)^{3x}[/tex] + 3

To find the value of given function at x= 0 , we need to substititute zero in place of x.

f(x) at x=0 will be [tex]-8(2)^{3(0)}[/tex] + 3

                        = [tex]-8(2)^{0}[/tex] + 3

                        = [tex]-8(1)[/tex] + 3  { we know that [tex]m^{0} = 1[/tex] }

                        = - 8 + 3

                        = -5

∴The value of function at x=0 is found to be -5

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Refer to the attachment for complete question

there are 407,230 possible coupon codes that can be generated using the given format.

To find the number of possible coupon codes, we need to count the total number of ways to choose three letters from 23, one digit from 10, and one letter from 23 (since we can repeat letters). Combinations

The number of ways to choose three letters from 23 is:  

23[tex]C_{3}[/tex] = (232221)/(321) = 1771    

The number of ways to choose one digit from 10 is simply 10.

The number of ways to choose one letter from 23 (allowing repetition) is 23.

Therefore, the total number of possible coupon codes is:

1771 * 10 * 23 = 407,230

So there are 407,230 possible coupon codes that can be generated using the given format.

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It will take 5.71 hours for Pam and Erin to be 14 miles apart.

If Pam and Erin start at the same point and rollerblade in different directions, then the total distance between them at any given time can be represented by the Pythagorean Theorem. The Pythagorean Theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

Let x represent the number of hours it takes for Pam and Erin to be 14 miles apart. Then, the distance traveled by Pam will be 2x, and the distance traveled by Erin will be 4x.

Using the Pythagorean Theorem, the equation that represents the distance between them can be written as 2x^2 + 4x^2 = 14^2.

Solving this equation, we get 6x^2 = 196, which can be written as x^2 = 32.75. Taking the square root of both sides, we get x = 5.71.

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The pair of supplementary angles in the given situation is:

136 + 44 = 180°

135 + 45 = 180°

154 + 26 = 180°

A supplementary angle is an angle that sums to 180 degrees.

For example, the 130° and 50° angles are complementary because the sum of 130° and 50° is 180°.

Similarly, the sum of the supplementary angles is 90 degrees.

An apex angle is an angle opposite at the intersection of two straight lines, and an adjacent angle is two angles next to each other.

So, the pair of supplementary angles would be:

136 + 44 = 180°

135 + 45 = 180°

154 + 26 = 180°

Therefore, the pair of supplementary angles in the given situation is:

136 + 44 = 180°

135 + 45 = 180°

154 + 26 = 180°

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Meg has 12 different options to choose from when it comes to buying a new pair of skates based on combinations.

Meg has three options to choose from - roller skates, figure skates, and hockey skates, and four options for the color of her skates - black, white, blue, and silver. To find the total number of different combinations, we have to multiply the number of options for each category.

Total number of combinations of skates that Meg can choose from is:

3 (types of skates) x 4 (colors of skates) = 12

Meg has 12 different options to choose from when it comes to buying a new pair of skates.

It is important to note that this calculation assumes that Meg can choose any combination of skate type and color. If certain types of skates only come in certain colors, or if certain colors are only available for certain types of skates, then the number of possible combinations will be less than 12. Additionally, other factors such as size and brand of the skates may also affect the total number of options available to Meg.

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The height of the triangle is approximately 7.31 units.

What is  Pythagorean theorem ?

The Pythagorean theorem is a fundamental theorem in geometry that relates to the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In mathematical notation, the Pythagorean theorem can be written as:

a^2 + b^2 = c^2

where a and b are the lengths of the legs (the sides adjacent to the right angle) and c is the length of the hypotenuse.

According to the question:

Since triangle ABC is a right triangle with angle B = 90 degrees, we can use the Pythagorean theorem to find the length of side BC:

[tex]BC^2 = AC^2 - AB^2[/tex]

[tex]BC^2 = 30^2 - h^2[/tex]

[tex]BC = \sqrt{30^2 - h^2}[/tex]

Now, let's consider triangle ABD. We know that AD = 25 and DC = 11, so BD = BC - DC:

BD = BC - DC

[tex]BD = \sqrt{30^2 - h^2} - 11[/tex]

Since the line passing through vertex A is perpendicular to BC, we know that triangles ABD and ABC are similar. Therefore, we can use the ratio of corresponding sides to find the value of h:

h/AB = AB/AC

h/AB = AB/30

[tex]AB^2 = h*30[/tex]

[tex]AB =\ sqrt{h*30}[/tex]

Now, using the fact that AD + DC = BC, we can write:

AD + DC = BD + AB

[tex]25 + 11 = \sqrt{30^2 - h^2} - 11 +\sqrt{h*30}[/tex]

[tex]36 = \sqrt{30^2 - h^2} + \sqrt{h*30}[/tex]

Squaring both sides, we get:

[tex]1296 = 30^2 - h^2 + 2\sqrt{h*30}\sqrt{30^2 - h^2} + h*30[/tex]

[tex]1296 = 900 - h^2 + 2\sqrt{30h - h^3} + 30*h[/tex]

[tex]396 = 32\sqrt{30*h - h^3}[/tex]

Squaring again, we get:

[tex]156816 = 960h^2 - 96h^4[/tex]

[tex]h^4 - 10h^2 + 1639/12 = 0[/tex]

Using the quadratic formula, we get:

[tex]h^2 = (10 \± \sqrt{10^2 - 4(1)(1639/12))}/2[/tex]

[tex]h^2 = (10 \± \sqrt{1561})/2[/tex]

Since h must be positive, we take the positive square root:

[tex]h = \sqrt{(10 + sqrt(1561)}/2) \approx 7.31[/tex]

Therefore, the height of the triangle is approximately 7.31 units.

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-------------------------------------------------------------------------------------------------------------

Answer:  Option A, [tex]\textsf{6a}^3\textsf{ + 22a}^2\textsf{ + 14a - 10}[/tex]

-------------------------------------------------------------------------------------------------------------

Given:  [tex]\textsf{3a + 5 and 2a}^2\textsf{ + 4a - 2}[/tex]

Find:  [tex]\textsf{The product of the two given equations}[/tex]

Solution: The first step toward solving this problem would be to distribute the 3a and 5 to each of the values in the second equation.

After doing so, we can simplify each of the expressions until we have one equation.  This can be done by both some simple algebra and combining of like terms.

Therefore, the correct answer to this question is Option A, [tex]\textsf{6a}^3\textsf{ + 22a}^2\textsf{ + 14a - 10}[/tex].

Answer:

Step-by-step explanation:

[tex]\sqrt{87x} =\sqrt{29\times3\times x}[/tex]

When [tex]x=21[/tex] we get:

    [tex]\sqrt{87x} =\sqrt{29\times3\times 21}[/tex]

              [tex]=\sqrt{29\times3\times3\times7}[/tex]

              [tex]=3\sqrt{29\times7}[/tex]

               [tex]=3\sqrt{203}[/tex]

So when [tex]x=21[/tex] we can simplify [tex]\sqrt{87x}[/tex].

Answer:

Since $\angle GHJ=90^\circ$, arc $GJ$ is a quarter of the circumference of circle $H$. The formula for the circumference of a circle is $C=2\pi r$, where $r$ is the radius, so the circumference of circle $H$ is:

$$C=2\pi \cdot 20 = 40\pi$$

Since arc $GJ$ is a quarter of the circumference, its length is:

$$\frac{1}{4} \cdot 40\pi = 10\pi$$

Therefore, the length of arc $GJ$ is $10\pi$ units.

The probability that a randomly selected box of cold medicine will cost more than $13 is 0.4542 or 45.42%.


Given the mean of the normally distributed cold medicine = $12.75 and the standard deviation = $2.15 and the random variable x, the probability of a randomly selected box of cold medicine costing more than $13 needs to be found.
Now, as we are given mean and standard deviation, we can standardize the normal distribution and then use the Z table or calculator to find the probability.
The formula for standardizing the normally distributed curve:
Z = (X - μ) / σ
where
Z = Standardized score
X = Score value
μ = Mean
σ = Standard deviation
Here, we have to find the probability of a randomly selected box of cold medicine will cost more than $13.
So, the formula becomes:
Z = (X - μ) / σ = ($13 - $12.75) / $2.15 = 0.116
Using the Z-table, the area to the left of Z = 0.116 is 0.5458
Thus, the probability that a randomly selected box of cold medicine will cost more than $13 is:
1 - 0.5458 = 0.4542 or 45.42%
Therefore, the probability of a randomly selected box of cold medicine costing more than $13 is 0.4542 or 45.42%.

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Using the trigonometric Identities, [tex]tan(A - B) =\frac{-5+12\sqrt{13} }{12+5\sqrt{13} }[/tex]

Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.

Given

[tex]sin(A) =\dfrac{5}{13}[/tex]

[tex]\dfrac{\pi }{2} < A < \pi[/tex]

Using the trigonometric identity

[tex]sin^2A+cos^2A=1[/tex]

[tex]cosA =\sqrt{1-sin^2A}[/tex]

[tex]cosA =\sqrt{1-(\dfrac{5}{13})^2 }[/tex]

[tex]cosA =-\dfrac{12}{13}[/tex]

[tex]tanA=\dfrac{sinA}{cosA}[/tex]

[tex]tanA =\dfrac{\frac{5}{13} }{\frac{-12}{13} }[/tex]

[tex]tanA =-\dfrac{5}{12}[/tex]

[tex]tan(A-B) =\dfrac{tanA-tanB}{1+tanAtanB}[/tex]

[tex]=\dfrac{-\frac{5}{12}-(-\sqrt{13}}{1+(-\frac{5}{12})(-\sqrt{3}) }[/tex]

[tex]=\dfrac{5-12\sqrt{13}}{-12-5\sqrt{3} }[/tex]

[tex]=\dfrac{-5+12\sqrt{13} }{12+5\sqrt{13} }[/tex]

Option D is correct.

Hence, [tex]tan(A - B) =\dfrac{-5+12\sqrt{13} }{12+5\sqrt{13} }[/tex]

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Therefore, the result of (u*v)v is a vector with components (80, -16).

In mathematics, a vector is a mathematical object that has both magnitude and direction. Vectors are used to represent physical quantities such as force, velocity, and acceleration, as well as abstract mathematical quantities such as points, lines, and planes. A vector is usually represented geometrically as an arrow in a coordinate system, where the length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector. Vectors can be added, subtracted, and multiplied by scalars, and these operations follow specific rules that make them a powerful tool for solving problems in mathematics, physics, engineering, and other fields. Vectors can have any number of dimensions, but in two-dimensional space, a vector can be represented as an ordered pair of numbers (x, y), where x and y represent the horizontal and vertical components of the vector, respectively. In three-dimensional space, a vector can be represented as an ordered triplet of numbers (x, y, z), and so on for higher dimensions.

Here,

To find (u*v)v, we first need to find the dot product of u and v, which is defined as:

u · v = (4, 4) · (-5, 1)

= (4 × -5) + (4 × 1)

= -20 + 4

= -16

Next, we need to multiply the dot product of u and v by v:

(u*v)v = (-16)(-5, 1)

= (80, -16)

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