The probability that each item coming off a production line is defective is p and the probability that it is non-defective is q, 0 < p < 1, p + q = 1. At the beginning of a day’s production, a quality control officer repeatedly inspects items each coming off a production line until he inspects n items. Let X be the number of defective items he finds. (a) Write down without proof the probability that X = k, indicating the possible values of k. Hence, by considering the expansion of 〖(p+q)〗^n+〖(q-p)〗^n. Show that the probability that X is even is 1/2[1+〖(1-2p)〗^n] .(Note that 0 is an even number) (b) Find the expected value of X, and write down without proof the variance of X. (c) If E(X) = 0.0125 and Var(X) = 0.9875, find to 4 decimal places the probability that X is odd.

Answer: [tex]\frac{216}{185} \;\approx 1.1676[/tex]

Given:

   [tex]\displaystyle \frac{\frac{12}{37}}{ \frac{5}{18} }[/tex]

Use the method of "keep, change, flip:"

* keep the first fraction, change to multiplication, flip the second

   [tex]\displaystyle \frac{12}{37}* \frac{18}{5}[/tex]

Multiply across and divide:

   [tex]\displaystyle \frac{216}{185} \;\approx 1.1676[/tex]

The area of the square with given side 18 cm is 324 square cm.

The equation for a square's surface area A square is a quadrilateral with four equal sides and four right angles, which is how Side x Side is obtained. By multiplying one side's length by the other side's length, which is likewise the same length, one may get the area of a square, which is the amount of space within the square. As a result, Side x Side is the formula for calculating a square's area.

Given that the length of the side of square = 18cm.

The area of the square is given as:

A = (s)(s)

Substituting the value we have:

Area = 324 square cm

Hence, the area of the square is 324 square cm.

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The solution to the given problem of speed comes out to be v=21.12m/s.

How quickly something is moving is measured by its speed at a distance. How far an object moves in one unit of time is determined by its speed. Speed is calculated as follows: speed = distance * time. The most widely used speed measurement units are meters per second (m/s), kilometers per hour (km/h), and miles per hour (mph) (mph).

Here,

Given :  A 2.0 m long cord is supporting a 3.3 kg lump of clay.

Two swing up to an absolute maximum of 0.080 meters

Ball and clay's subsequent impact velocity

=>√(2*10*0.08)=1.264m/s

To find the velocity of the ball before collision

0.21*v=(3.3+0.21)*1.264

v=21.12m/s

Therefore, the solution to the given problem of speed comes out to be v=21.12m/s.

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In  323,400 ways it can be done by using the Combination formula.

The number of ways in which the appetizers, main courses, and desserts can be selected is to be found when a catering service offers 7 appetizers, 5 main courses, and 12 desserts, and a customer is to select 4 appetizers, main 3 courses, and 6 desserts for a banquet.

Let's find the number of ways to choose 4 appetizers from the 7 available:

= ⁷C₄ ⇒ 35 ways (using combinations).

Let's find the number of ways to choose 3 main courses from the 5 available:

= ⁵C₃ ⇒ 10 ways (using combinations).

Let's find the number of ways to choose 6 desserts from the 12 available:

= ¹²C₆ ⇒ 924 ways. (using combinations).

Therefore, the total number of ways in which the customer can select 4 appetizers, 3 main courses, and 6 desserts is:

35 × 10 × 924 ⇒ 323,400 ways.

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The zeros of the polynomial function f(x) are 3, -[(2 -i√2) / 3] and  -[(2 + i√2) / 3]

To find the zeros of the polynomial function f(x), we need to find the values of x for which f(x) = 0.

We can start by factoring out a common factor of 3x^2 from the polynomial:

f(x) = 3x^3 - 5x^2 - 10x - 6

f(x) = 3x^2(x - 5/3) - 2(5x + 3)

Now, we can set each factor equal to zero and solve for x:

3x^2(x - 5/3) - 2(5x + 3) = 0

3x^3 - 5x^2 - 10x - 6 = 0

x = 3, -[(2 -i√2) / 3],  -[(2 + i√2) / 3]

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The value of length SP for the two given similar triangles is 11.

Similar triangles are triangles that have the same shape but are different in size. In other words, their corresponding angles are equal, and their corresponding sides are proportional.

This means that if you were to take one of the similar triangles and enlarge or shrink it, while keeping the angles the same, it would still be a similar triangle.

The value of length SP is calculated by applying the following method;

|SP| = |ST|

3x + 2 = 5x - 4

3x - 5x = -4 - 2

-2x = - 6

x = 3

Length SP = 3x + 2

                 = 3(3) + 2

                 = 11

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The distance between these two streets on Broad St 0.2 miles.

We must apply the idea of comparable triangles to this issue in order to find a solution. Assume that Thomas St. and Denny Way. are separated by x miles on Broad St. Then, we can establish the ratio shown below:

0.2 miles on Aurora Avenue equals x miles on Broad Street

By cross-multiplying and simplifying, we may find the value of x:

Distance on Aurora Ave / (x * 0.2 miles) on Broad St

Broad Street distance is equal to (x * 0.2 miles)/0.2 miles. (since the distance on Aurora Ave is given as 0.2 miles)

Broad Street: distance = x

As a result, Thomas St. and Denny Way are separated by x miles, or 0.2 miles, on Broad St. Thus, the response is:

Distance between Thomas St and Denny Way on Broad St = 0.2 miles (rounded to the nearest tenth of a mile)

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The graph's scales could be represented by the x-axis, which could be labelled from 0 to 12 with a scale of 1, and the y-axis, which could be labelled from 0 to 100 with a scale of 10.

The horizontal scale used in a graph is called the X axis. Measurements on the coordinate plane are made using it as a reference line. The position of an object on that plane is described by its distance from the x and y axes.

All potential data that can be graphed is represented by the numbers on the X and Y axes. Each number here is different by ten. Scale is the name for this. On a coordinate grid, scale is the distance between each square.

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The area of the given trapezoid above would be = 1.2m²

A trapezoid is defined as a quadrilateral that has four sides with a pair of parallel sides.

To calculate the area of the trapezoid the formula below should be used. That is;

Area = 1/2 (a+b) h

where;

a= 1.7m

b = 0.7m

h = sin∅ = opposite/hypotenuse

where;

opposite = ?

hypotenuse =1.4

sin 45° = h/1.4

h= 0.707106781 ×1.4

h = 1m

Area= 1/2 (1.7+0.7) × 1

= 1.2 m²

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The radii of circle are AB, AC and AE.

The radius of a circle is the distance from the center of the circle to any point on its circumference. To find the radius of a circle, you can use the formula:

radius = diameter / 2

where the diameter is the distance across the circle, passing through its center.

Here, in this figure we need to identify the radii. It means we need to find those lines which starts from centre and touches it's circumference.

Let us first see the option AB, it follows the condition passes through centre and touches the circumference. So, it is radius of circle.

Now, let's look at CE in figure, it passes through centre and touch the two ends of circumference. So, it is diameter not radius.

Now, AC satisfies the condition of radius. So, AC is radius.

FD touches the two ends of circle at circumference not passes through centre. So, it is chord not radius.

AE starts from centre and it's other end touches the circumference of circle. So, it is radius.

So, the radii of circle in given figure are : AB, AC and AE.

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A linear equation to represent the line shown on the graph is given as:

y = 2x - 2.

General point-slope form is:

y = mx + c

(x1, y1) = (3, 2)

y intercept = -2

m = (2 + 2)/(2 - 0)

m = 2

y = 2x + (-2)

y = 2x - 2

Thus,  a linear equation to represent the line shown on the graph is given as: y = 2x - 2.

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The correct question is-

Write a linear equation to represent the line shown on the graphs shown by question 7.

Aiden's estimated the length of a piece of rope as 8.5 inches, while its actual length is 7.1 inches. Therefore, the percent error of Aiden's estimate is 11.3%.

To calculate the percent error, you first need to find the difference between the actual length and the estimated length. Subtract 7.1 inches from 8.5 inches and you get 1.4 inches. This is the difference between the two lengths.
Next, divide the difference by the actual length and multiply by 100. The equation is: (difference/actual length) * 100. So, (1.4/7.1)*100 = 11.3%. Therefore, the percent error of Aiden's estimate is 11.3%.
It is important to be accurate when making measurements and estimates. A small difference in numbers can lead to a large error in the final result. Knowing the percent error can help you to improve your measurements and estimates and achieve greater accuracy.

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a)  The probability of finding a value greater than $24 is given by:P(X > $24) = P(Z > (24 - 21) / 4) = P(Z > 0.75)Using the standard normal distribution table, we can find that P(Z > 0.75) = 0.2266.Rounding this result to four decimal places, we have:P(X > $24) = 0.2266.

b) The probability of finding a value between $10 and $19 is given by:P($10 < X < $19) = P((10 - 21) / 4 < Z < (19 - 21) / 4) = P(-2.75 < Z < -0.5)Using the standard normal distribution table, we can find that P(-2.75 < Z < -0.5) = P(Z < -0.5) - P(Z < -2.75) = 0.3085 - 0.0030 = 0.3055.Rounding this result to four decimal places, we have:P($10 < X < $19) = 0.3055.

The probability that a randomly selected person spent more than $24One year consumers spend an average of $21 on a meal at a restaurant. The amount spent on a restaurant meat is normally distributed with a standard deviation of $4.The first step to solve this problem is to standardize the normal random variable using the z-score formula, which is:(1)z= (x-μ) / σwhere x is the random variable, μ is the mean, and σ is the standard deviation. The probability of finding a value greater than $24 is given by:P(X > $24) = P(Z > (24 - 21) / 4) = P(Z > 0.75)Using the standard normal distribution table, we can find that P(Z > 0.75) = 0.2266.Rounding this result to four decimal places, we have:P(X > $24) = 0.2266.

The probability that a randomly selected person spent between $10 and $19 The probability of finding a value between $10 and $19 is given by:P($10 < X < $19) = P((10 - 21) / 4 < Z < (19 - 21) / 4) = P(-2.75 < Z < -0.5)Using the standard normal distribution table, we can find that P(-2.75 < Z < -0.5) = P(Z < -0.5) - P(Z < -2.75) = 0.3085 - 0.0030 = 0.3055.Rounding this result to four decimal places, we have:P($10 < X < $19) = 0.3055.

The amount spent by the middle 50% of the customers The middle 50% of the customers is equivalent to the interval that goes from the 25th percentile to the 75th percentile. This interval is also known as the interquartile range (IQR).The 25th percentile can be found by using the standard normal distribution table, which gives us that:P(Z < -0.6745) = 0.25 Solving for Z, we have:Z = -0.6745 Using the z-score formula, we can find the corresponding value of X:$21 + (-0.6745)($4) = $17.30

Therefore, the lower limit of the IQR is $17.30.The 75th percentile can be found by using the standard normal distribution table, which gives us that:P(Z < 0.6745) = 0.75 Solving for Z, we have:Z = 0.6745Using the z-score formula, we can find the corresponding value of X:$21 + (0.6745)($4) = $24.70

Therefore, the upper limit of the IQR is $24.70.The amount spent by the middle 50% of the customers is between $17.30 and $24.70.

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Problem 1:

a) The probability that no customers will be sent to another hotel is 0.039.

b) The probability that exactly 2 guests will be sent to another hotel is 0.228.

c) The probability that 3 or more guests will be sent to another hotel is 0.492.

Problem 2:

a) ( ≤ 20) - Probability of having 20 or fewer customers show up for the reservations.

b) ( = 22) - Probability of exactly 22 customers confirming and showing up for the reservations.

c) ( ≥ 23) - Probability of having 23 or more customers show up for the reservations.

Problem 1:

a. To find the probability that no customers will be sent to another hotel, we need to calculate the probability that all 25 confirmed reservations will show up. Since the hotel manager estimates that 15% of reservations are "no-shows",

Then the probability that a reservation will show up is 1 - 0.15 = 0.85. The probability that all 25 guests will show up is,

P(all 25 show up) = [tex](0.85)^{25}[/tex]

                             = 0.039

So the probability that no customers will be sent to another hotel is 0.039.

b. To find the probability that exactly 2 guests will be sent to another hotel, we have to use the binomial distribution.

The probability of a reservation being a no-show is 0.15, and the probability of a reservation showing up is 0.85. We have 25 confirmed reservations, so the probability of exactly 2 no-shows is,

P(exactly 2 no-shows) = (25 choose 2)[tex](0.15)^2 (0.85)^{23}[/tex]

                                     = 0.228

So the probability that exactly 2 guests will be sent to another hotel is 0.228.

c. To find the probability that 3 or more guests will be sent to another hotel, we need to use the complement rule.

The probability of 0, 1, or 2 guests being sent to another hotel is,

⇒ P(0 guests sent) + P(1 guest sent) + P(2 guests sent)

   = [tex](0.85)^{25}[/tex]+ (25 choose 1)[tex](0.15)^1 (0.85)^{24}[/tex] + (25 choose 2)[tex](0.15)^2 (0.85)^{23}[/tex]

   = 0.039 + 0.168 + 0.301

   = 0.508

The probability of 3 or more guests being sent to another hotel is,

P(3 or more guests sent) = 1 - P(0, 1, or 2 guests sent)

                                         = 1 - 0.508

                                         = 0.492

So the probability that 3 or more guests will be sent to another hotel is 0.492.

Problem 2:

a) To find ( ≤ 20),

We have to add up the probabilities of all the possible values of from 0 to 20. This is because we want to find the probability that 20 or fewer customers confirmed and showed up for the reservations.

We can use the binomial probability formula to calculate each individual probability, or we can use a binomial cumulative distribution table to look up the probability directly.

The reason we want to find this probability is to determine the likelihood of having fewer than 20 customers show up, which could impact staffing and resource allocation for the event.

b) To find ( = 22),

Use the binomial probability formula to calculate the probability of exactly 22 customers confirming and showing up for the reservations. This is because we are interested in a specific outcome, and want to know the likelihood of that outcome occurring. Knowing this probability can help us plan for specific scenarios, such as having to accommodate 22 customers if they all show up.

c) To find ( ≥ 23),

We have to add up the probabilities of all the possible values of from 23 to 25. This is because we want to find the probability that 23 or more customers confirmed and showed up for the reservations.

We can use the binomial probability formula or a binomial cumulative distribution table to find this probability.

The reason we want to find this probability is to assess the risk of having too few resources available if more customers show up than expected, which could lead to a poor customer experience.

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Measure of ∠1 = 126 deg, Measure of ∠4 = 54 deg, and measure of ∠7 = 126 deg.

∵ Lines p and q are parallel, and t is transversal,

∴ ∠3 and ∠7 form pair of corresponding angles,

⇒ ∠3 and ∠7 are equal

∴∠7=126 deg.

Also, ∠3 and ∠1 are vertically opposite angles,

∴ ∠3 and ∠1 are equal.

⇒ ∠1=126

Again, as t is a straight line and line p intersects it,

∴ ∠3 and ∠4 form linear pair.

⇒ ∠3 and ∠4 are complementary.

⇒ ∠3+∠4=180

⇒ ∠4+126=180

⇒ ∠4=180-126

⇒ ∠4=54

Hence, measure of ∠1 is 126 deg, that of ∠4 is 54 deg, and that of ∠7 is 126 deg.

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[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ P(\stackrel{x_1}{19}~,~\stackrel{y_1}{10})\qquad Q(\stackrel{x_2}{26}~,~\stackrel{y_2}{13})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ PQ=\sqrt{(~~26 - 19~~)^2 + (~~13 - 10~~)^2} \implies PQ=\sqrt{( 7 )^2 + ( 3 )^2} \\\\\\ PQ=\sqrt{ 49 + 9 } \implies PQ=\sqrt{ 58 }\implies PQ\approx 7.62 \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ P(\stackrel{x_1}{19}~,~\stackrel{y_1}{10})\qquad Q(\stackrel{x_2}{26}~,~\stackrel{y_2}{13}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ 26 +19}{2}~~~ ,~~~ \cfrac{ 13 +10}{2} \right) \implies \left(\cfrac{ 45 }{2}~~~ ,~~~ \cfrac{ 23 }{2} \right)\implies \stackrel{ \textit{\LARGE M} }{\left(22\frac{1}{2}~~,~~11\frac{1}{2} \right)}[/tex]

Answer:

c = 18

Step-by-step explanation:

expand the factor in j(x) and compare like terms with k(x)

j(x) = 2(x + 3)² ← expand factor using FOIL

    = 2(x² + 6x + 9) ← distribute parenthesis by 2

   = 2x² + 12x + 18

compare to k(x) = 2x² + 12x + c

the 2 expressions are equivalent when c = 18

Answer:

Let x be the length of the base of the triangle, then the height h is given by h = 2x - 3 (since the height is 3 inches less than twice the length of the base).

The area of a triangle is given by the formula A = (1/2)bh, where b is the base and h is the height. We are given that the total area of the triangle is 7 square inches, so we can write:

(1/2)(x)(2x - 3) = 7

Multiplying both sides by 2 to eliminate the fraction, we get:

x(2x - 3) = 14

Expanding the left side, we get:

2x^2 - 3x = 14

Subtracting 14 from both sides, we get:

2x^2 - 3x - 14 = 0

We can now use the quadratic formula to solve for x:

x = (-b ± sqrt(b^2 - 4ac))/(2a)

where a = 2, b = -3, and c = -14. Plugging in these values, we get:

x = (-(-3) ± sqrt((-3)^2 - 4(2)(-14)))/(2(2))

= (3 ± sqrt(169))/4

= (3 ± 13)/4

Taking the positive value for x (since the length of the base must be positive), we get:

x = (3 + 13)/4

= 4

Therefore, the length of the base is 4 inches. To find the height h, we can use the formula h = 2x - 3:

h = 2(4) - 3

= 5

So the height of the triangle is 5 inches.

The correct sequence of transformations is: reflect across the y-axis and then translate 9 units to the right.

In everyday parlance, transformation refers to a mathematical function. A transformation is defined as the invertible function from any set X to its own set X or any other set Y. The transformation for any term may merely signal that the geometric component of this particular function is being studied.

The correct sequence of transformations that could have been used to transform triangle ABC to produce triangle A′′B′′C′′ is:

Triangle ABC was reflected across the y-axis and then translated 9 units right.

To see why, let's compare the coordinates of the corresponding vertices of both triangles:

A (-6, 2) ---> A'' (6, -5)

B (-3, 6) ---> B'' (3, -1)

C (-3, 2) ---> C'' (3, -5)

If we reflect triangle ABC across the y-axis, we obtain a new triangle A'B'C' with vertices:

A' (6, 2)

B' (3, 6)

C' (3, 2)

Then, if we translate triangle A'B'C' 9 units to the right, we obtain triangle A''B''C'':

A'' (6+9, 2) = (15, 2)

B'' (3+9, 6) = (12, 6)

C'' (3+9, 2) = (12, 2)

Which has the same coordinates as triangle A''B''C'' given in the problem statement. Therefore, the correct sequence of transformations is: reflect across the y-axis and then translate 9 units to the right.

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Answer:

x = - 4 , x = 4

Step-by-step explanation:

the expression is undefined if the denominator equals zero

equate the denominator to zero and solve for x

x² - 16 = 0 ( add 16 to both sides )

x² = 16 ( take square root of both sides )

x = ± [tex]\sqrt{16}[/tex] = ± 4

that is the expression is undefined when x = - 4 or x = 4

the height VO of the rectangular pyramid VPQRS is 5 cm.

WHAT IS RECTANGULAR PYRAMID?

A rectangular pyramid is a type of pyramid where the base is a rectangle and the lateral faces are triangles with a common vertex (apex) that is not in the plane of the base. It is a polyhedron with a rectangular base and triangular faces that meet at a single vertex. The height of the pyramid is the perpendicular distance from the apex to the base. The volume of a rectangular pyramid can be calculated using the formula:

V = (1/3) * base area * height

where base area is the area of the rectangular base and height is the perpendicular distance from the apex to the base.

To find the height VO of the rectangular pyramid VPQRS, we can use the formula for the volume of a pyramid:

V = (1/3) * base area * height

where base area is the area of the rectangle formed by the base of the pyramid, and height is the height of the pyramid.

We are given that the volume of the pyramid is 100 cm³. We can also find the base area by multiplying the length PQ by the width QR:

base area = PQ * QR = 10 cm * 6 cm = 60 cm²

Substituting these values into the formula for the volume of a pyramid, we get:

100 cm³ = (1/3) * 60 cm² * height

Simplifying, we get:

height = (100 cm³ * 3) / (60 cm²)

height = 5 cm

Therefore, the height VO of the rectangular pyramid VPQRS is 5 cm.

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The volume of the rectangular prism in cubic feet is solved to be

The volume of a rectangular prism is given by the formula

V = l x w x h,

where

l, w, and h represent the length, width, and height of the prism, respectively.

in the problem, the dimensions are:

Therefore, the volume is:

= 1.5 feet * 3.5 feet * 2 feet

= 10.5 feet³

That is to say the volume of the prism is 10.5 feet³

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Answer:

What do you mean?

Step-by-step explanation:

Some questions are new and have not been answered yet.

The result would be (C) 27 if the above statement is accurate.

Three categories of integers exist: Zero (0) (0) Good integers (Natural numbers) Integer Negatives (Additive inverse of Natural Numbers).

As the two numbers are consecutive, we will refer to the smaller integer as "x" and the larger one as "x + 1".

In accordance with the issue, we have:

2x + (1/2)(x + 1) = 33

To eliminate the fraction, multiply everything by 2 and you obtain the following:

4x + x + 1 = 66

If we simplify, we get:

5x = 65

When we multiply both parts with 5, we get:

x = 13

Hence, 13 is the smaller number while 14 is the larger number.

The two integers' total is:

13 + 14 = 27

Hence, the response is (C) 27.

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Answer:

Trinominal

Step-by-step explanation:

By definition, Trinominals are those expressions having 3 values, in this case, x^2, x, and the constant 6 are the values.

hope it helps.

Step-by-step explanation:

To find C and D, we can use the given information about the initial position of the person on the ride.

When the person starts the ride, they are 10 feet off the ground. This means that the cosine function has a vertical shift of 10 units, so we have:

f(x) = Acos(Bx - C) + D = 60cos(π/2x - C) + D

At the start of the ride, when x = 0, f(x) = 10. Substituting these values, we get:

10 = 60cos(-C) + D

Simplifying, we get:

D = 10 - 60cos(-C)

We can also use the fact that the minimum height of the ride is 4 feet above the ground. This means that the cosine function has a vertical shift of 4 units, so we have:

f(x) = Acos(Bx - C) + D = 60cos(π/2x - C) + D

At the lowest point of the ride, when x = 1/4, f(x) = 4. Substituting these values, we get:

4 = 60cos(π/8 - C) + D

Substituting D = 10 - 60cos(-C) from the first equation, we get:

4 = 60cos(π/8 - C) + 10 - 60cos(-C)

Simplifying, we get:

cos(-C) = (4 - 10 - 60cos(π/8 - C))/(-60)

cos(-C) = (3cos(π/8 - C) - 1)/2

Using the identity cos(-x) = cos(x), we can rewrite this as:

cos(C) = (3cos(π/8 - C) - 1)/2

Solving for C numerically, we get:

C ≈ 0.438

Substituting this value of C and D = 10 - 60cos(-C) into the equation for f(x), we get:

f(x) = 60cos(π/2x - 0.438) + 10 + 60cos(0.438)

The value of angle E and angle F are both 22.8°

Circle theorem includes the concept of tangents, sectors, angles, the chord of a circle and proofs. A circle is the locus of all points in a plane which are equidistant from a fixed point.

The theorem that states that the angle at the center is twice the angle at the circumference is applied in this scenario.

angle P = angle of arc DG = 45.6°

therefore angle E = 1/2 angle of arc DG

= 1/2 × 45.6

= 22.8°

angle E = angle F = 22.8° . This because angle in the same segment are equal.

therefore angle E = angle F = 22.8°

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Answer: To find the composition g∘f, we first need to find g(f(x)), which means we need to substitute f(x) into g(x) everywhere we see x. So we have:

g(f(x)) = g(2x+5) = (2x+5)^2 - 3

Expanding the square, we get:

g(f(x)) = (4x^2 + 20x + 25) - 3

Simplifying, we get:

g(f(x)) = 4x^2 + 20x + 22

Therefore, the composition g∘f is equal to 4x^2 + 20x + 22.

Step-by-step explanation: