1. find a formula for the sum 1 1.2 1 2.3 ... 1 n(n 1) by examining the values of this expression for small values of n prove the formula you conjectured in first part by induction.

The integral. 1 [*+1590 Joe (x15 + 15*)dx is  154/3.

Based on the information provided, I assume you want to evaluate the integral of a given function. Let me rewrite the function in a more standard format:

∫(x^15 + 15x) dx

Now, let's evaluate the integral step by step:

1. Identify the function within the integral: f(x) = x^15 + 15x

2. Apply the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n is a constant and C is the constant of integration.

3. Using the power rule for the first term: ∫x^15 dx = (x^(15+1))/(15+1) = (x^16)/16

4. Using the power rule for the second term: ∫15x dx = 15∫x dx = 15(x^2)/2

5. Combine both terms and add the constant of integration, C: (x^16)/16 + (15x^2)/2 + C

So, the evaluated integral of the given function is:
Putting these together, we have:

∫[1,5] (x^2 + 15) dx = [(1/3)x^3 + 15x] evaluated from x=1 to x=5

Plugging in these values, we get:

[(1/3)(5^3) + 15(5)] - [(1/3)(1^3) + 15(1)]

Simplifying, we get:

(125/3 + 75) - (1/3 + 15)

= (200/3) - (46/3)

= 154/3
= (x^16)/16 + (15x^2)/2 + C

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The completed relationship is:

0.001 grams = 0.0000000551 µg

To complete the relationship, we need to convert the units of the left-hand side and simplify the right-hand side.

Starting with the left-hand side:

1 mg = 1 milligram = 0.001 grams

Now, we can substitute this into the relationship:

For the right-hand side:

1100 in binary is equal to[tex]12^3 + 12^2 + 02^1 + 02^0 = 8 + 4 = 12[/tex]

10001000 in binary is equal to [tex]12^7 + 02^6 + 02^5 + 02^4 + 12^3 + 02^2 + 02^1 + 02^0 = 128 + 8 = 136[/tex]

100 in binary is equal to [tex]12^2 + 02^1 + 0*2^0 = 4[/tex]

Now, we can substitute these values into the relationship:

0.001 grams = 12 µg / 136 / 4

Simplifying the right-hand side:

12 µg / 136 / 4 = 12 * (1/1000000) * (1/136) * (1/4) = 0.00000005514706...

Rounding this to a reasonable number of significant digits, we get:

0.0000000551

Therefore, the completed relationship is:

0.001 grams = 0.0000000551 µg

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Answer:

c

Step-by-step explanation:

Answer:

Step-by-step explanation:

Pictograph

Pictograms are a powerful tool for visualizing data and are widely used in a variety of different fields, from marketing and advertising to science and education.

A pictogram is a type of chart that uses symbols instead of words or numbers to portray data. Pictograms are often used in data visualization and are particularly useful for presenting complex information in a simple and easily understandable way. Pictograms can be used to represent a wide range of data, including statistical information, demographic data, and geographical information. They are also commonly used in advertising and marketing, as they are a powerful tool for communicating ideas and concepts quickly and effectively. Pictograms can be created using a variety of different techniques, including hand-drawn illustrations, computer-generated graphics, and photographs. They are typically presented in a grid format, with each symbol representing a single data point. Pictograms can be used to show trends, compare data sets, and highlight key points. They are also a great way to make data more engaging and interactive, as users can explore the data by clicking on individual symbols or groups of symbols.

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Answer:

it's the natural error that exists between a sample and it's corresponding population

The 6th term of the geometric sequence for the first sequence is -15625.

The 6th term of the geometric sequence for the second sequence is -762.875

The common ratio of the sequence is found by dividing any term by its preceding term.

For the first sequence:

Common ratio = (-5) / (-1) = 5

To find the 6th term, we can use the formula for the nth term of a geometric sequence:

a_n = a_1 * r^(n-1)

where a_1 is the first term, r is the common ratio, and n is the term we want to find.

For the first sequence, we have:

a_1 = -1

r = 5

n = 6

a_6 = (-1) * 5^(6-1) = -15625

So the 6th term of the first sequence is -15625.

For the second sequence:

Common ratio = (-7) / (-2) = 3.5

Using the same formula, we have:

a_1 = -2

r = 3.5

n = 6

a_6 = (-2) * 3.5^(6-1) = -762.875

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The finance charge, based on the annual interest rate of 18 % would be $ 9. 07.

You need to find the periodic rate :

=  18 %  / 365

=  0. 04931506849315

Then the average daily balance :

Days 1 - 7 : $ 800 balance

Days 8 - 15 : $ 800 + $ 600 = $ 1400 balance

Days 16 - 20 : $ 1400 - $ 1000 = $ 400 balance

This allows us to find the average daily balance :
= ( ( 800 x 7 ) + ( 1, 400 x 8 ) + ( 400 x 5 ) ) / 20

= $ 940

The finance charge is:

= 940 x 0. 04931506849315 x 20

= $ 9. 07

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The sample mean of the population is 3/4 and the variance is 3/80. Using the central limit theorem, P([tex]\bar{X}[/tex] > 0.8) can be simplified as 0.003.

The mean of the population can be computed as follows:

µ = ∫x f(x) dx from 0 to 1

= ∫x (3x²) dx from 0 to 1

= 3/4

The variance of the population can be computed as follows:

σ² = ∫(x-µ)² f(x) dx from 0 to 1

= ∫(x-(3/4))² (3x²) dx from 0 to 1

= 3/80

By the Central Limit Theorem, as the sample size n = 80 is large, the distribution of the sample mean [tex]\bar{X}[/tex] can be approximated by a normal distribution with mean µ and variance σ²/n.

Therefore, P([tex]\bar{X}[/tex] > 0.8) can be approximated by P(Z > (0.8-0.75)/(sqrt(3/80)/sqrt(80))), where Z is a standard normal random variable.

Simplifying, we get P([tex]\bar{X}[/tex] > 0.8) ≈ P(Z > 2.73) ≈ 0.003.

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sin x = ± sqrt((1/2) (1 - cos^2x)) is the desired identity, which relates sin x to cos x.

To obtain the identity, we start by squaring both sides of the formula:

(sin x)^2 = (1/2i)^2 (e^ix - e^-ix)^2

Expanding the right-hand side using the binomial formula, we get:

(sin x)^2 = (1/4) (e^2ix - 2 + e^-2ix)

Next, we can use the identity e^ix e^-ix = 1 to simplify the expression:

(sin x)^2 = (1/4) (2cos^2x - 2)

Factoring out the 2, we get:

(sin x)^2 = (1/2) (1 - cos^2x)

Taking the square root of both sides, we obtain:

sin x = ± sqrt((1/2) (1 - cos^2x))

This is the desired identity, which relates sin x to cos x.

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The correct graph of function g(x) is shown in Option A.

We have o given that;

Function is,

⇒ g (x) = ax² + bx + c

Where, a, b and c are negative constants.

Since, In the given function all the coefficients are negative.

Hence, The equation of parabola is make down to x - axis.

Thus, The correct graph of function g(x) is shown in Option A.

So, Option A is true.

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Top-down processing refers to analysis that travels from the whole to the pieces, whereas bottom-up processing refers to analysis that does the opposite. Option A is Correct.

Making meaning of stimuli may be done in two distinct ways: bottom-up and top-down. In bottom-up processing, we let the input itself, free of any prior notions, influence our vision. In top-down processing, we interpret what we perceive by drawing on our prior knowledge and expectations.

According to the theory of "top down processing," our brains first build a notion of the overall picture based on prior knowledge before breaking it down into more detailed information. We use our perceptual set—past experiences, expectations, and emotions—to interpret the world around us. Option A is Correct.

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The numbers on the second die are [tex]0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6.[/tex]

Given that,

The sum of the two numbers will be [tex]1, 2, ..., 12[/tex] with an equal probability of [tex]\dfrac{1}{12}[/tex] each.

For the numbers on the second die, determine a set of numbers that, when combined with the numbers on the first die, will result in an equal probability for each possible sum from 1 to 12.

Let's analyze the possible sums and their corresponding combinations of numbers from the two dice:

Sum 1: The only combination that results in a sum of 1 is (1, 1).

Therefore, the second die should have a side with 1.

Sum 2: The combinations that result in a sum of 2 are (1, 1) and (2, 0).

Since we already have 1 on the second die, the other number should be 1.

Hence, the second die should have a side with 1 and a side with 0.

Sum 3: The combinations that result in a sum of 3 are (1, 2) and (2, 1).

Since we already have 1 on the second die, the other number should be 2.

Hence, the second die should have a side with 1 and a side with 2.

Following this pattern, determine the numbers on the second die as:

Sum 1: 1

Sum 2: 0, 1

Sum 3: 1, 2

Sum 4: 1, 3

Sum 5: 2, 3

Sum 6: 2, 4

Sum 7: 3, 4

Sum 8: 3, 5

Sum 9: 4, 5

Sum 10: 4, 6

Sum 11: 5, 6

Sum 12: 6

Thus, The numbers on the second die are [tex]0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6.[/tex]

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Answer:

This is the probability density function of X.

Step-by-step explanation:

To find the PDF of X, we need to find the probability of each possible value of X.

When two dice are thrown, the possible outcomes are the integers from 2 to 12, with each outcome having an equal probability of 1/36. Therefore, the PDF of X can be represented as follows:

P(X = 2) = 1/36

P(X = 3) = 2/36

P(X = 4) = 3/36

P(X = 5) = 4/36

P(X = 6) = 5/36

P(X = 7) = 6/36

P(X = 8) = 5/36

P(X = 9) = 4/36

P(X = 10) = 3/36

P(X = 11) = 2/36

P(X = 12) = 1/36

This is the probability density function of X.

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Answer:

To determine which graph represents the rational function f(x) = (x^2 - 16)/(x^2 - 2x - 8), we can analyze the behavior of the function as x approaches infinity and negative infinity, as well as the location and behavior of any vertical asymptotes, horizontal asymptotes, x-intercepts, and y-intercepts.

First, let's factor the denominator of the rational function:

x^2 - 2x - 8 = (x - 4)(x + 2)

Therefore, the rational function can be written as:

f(x) = (x^2 - 16)/((x - 4)(x + 2))

To find any vertical asymptotes, we need to look for values of x that make the denominator of the rational function equal to zero. Since the denominator is a product of two linear factors, the values that make it equal to zero are x = 4 and x = -2. Therefore, the rational function has vertical asymptotes at x = 4 and x = -2.

To find any horizontal asymptotes, we can look at the behavior of the function as x approaches infinity and negative infinity. As x becomes very large (either positive or negative), the highest degree term in the numerator and denominator of the rational function will dominate the expression. In this case, both the numerator and denominator have a highest degree of x^2, so we can apply the horizontal asymptote rule and divide the leading coefficient of the numerator by the leading coefficient of the denominator. This gives us:

y = 1

Therefore, the rational function has a horizontal asymptote at y = 1.

To find any x-intercepts, we need to look for values of x that make the numerator of the rational function equal to zero. Since the numerator is a difference of two squares, we can factor it as:

x^2 - 16 = (x - 4)(x + 4)

Therefore, the rational function has x-intercepts at x = 4 and x = -4.

To find the y-intercept, we can set x = 0 in the rational function:

f(0) = (-16)/(-8) = 2

Therefore, the rational function has a y-intercept at y = 2.

Based on this information, we can sketch the graph of the rational function as follows:

Out of the provided graphs, only graph (C) matches this description. Therefore, graph (C) represents the rational function f(x) = (x^2 - 16)/(x^2 - 2x - 8).

Answer: C

Step-by-step explanation:

The dimensions of the rectangular prism-shaped birdhouse are 742 inches, 4 inches, and 746 inches.

We have,

Let's first convert the volume of the birdhouse from cubic feet to cubic inches, since the dimensions are given in inches.

1 cubic foot = 12 inches x 12 inches x 12 inches = 1728 cubic inches

The volume of the birdhouse in cubic inches is:

= 128 cubic feet x 1728 cubic inches per cubic foot

= 221184 cubic inches

Let's represent the width of the birdhouse in inches as w, and its height

as h.

We know that the depth is 4 inches, and the height is 4 inches greater than the width.

Width = w inches

Depth = 4 inches

Height = w + 4 inches

The volume of the birdhouse is given by the formula:

Volume = Width x Depth x Height

Substituting the given values, we get:

221184 cubic inches = w x 4 inches x (w + 4) inches

Simplifying and rearranging, we get:

w + 4w - 55296 = 0

Using the quadratic formula, we find:

w = (-4 ± √(4² - 4(1)(-55296))) / (2(1))

w = (-4 ± √(2220800)) / 2

w = (-4 ± 1488) / 2

Since the width must be positive, we discard the negative solution and get:

w = (-4 + 1488) / 2

w = 742

Therefore,

The dimensions of the rectangular prism-shaped birdhouse are 742 inches, 4 inches, and 746 inches.

The width of the birdhouse is 742 inches, the depth is 4 inches, and the height is w + 4 = 746 inches.

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The vector representing the speed and direction of the motorboat is approximately 20.22 mph at an angle of 8.53 degrees with respect to the original direction of the boat.

To find the vector representing the speed and direction of the motorboat, we need to use vector addition. Let the velocity of the boat in still water be Vb and the velocity of the current be Vc. Then, the resulting velocity of the boat relative to the ground is Vr = Vb + Vc.
Since the boat is heading directly across the river, the velocity of the current is perpendicular to the direction of the boat. This means that we can use the Pythagorean theorem to find the magnitude of Vr:
|Vr|^2 = |Vb|^2 + |Vc|^2
|Vr|^2 = (20 mph)^2 + (3 mph)^2
|Vr|^2 = 409
|Vr| ≈ 20.22 mph
To find the direction of Vr, we can use trigonometry. Let θ be the angle between Vr and Vb. Then:
tan(θ) = |Vc| / |Vb|
tan(θ) = 3 / 20
θ ≈ 8.53 degrees

The true speed of the motorboat is simply the magnitude of Vb:  |Vb| = 20 mph

To find the vector representing the speed and direction of the motorboat, we need to consider both the motorboat's speed in still water and the river current's speed.
Step 1: Identify the motorboat's speed in still water (20 mph) and the river current's speed (3 mph).
Step 2: Represent the motorboat's speed as a vector. Since it is heading directly across the river, we can represent it as a horizontal vector: V_motorboat = <20, 0>.
Step 3: Represent the river current's speed as a vector. The current flows perpendicular to the motorboat's direction, so we can represent it as a vertical vector: V_current = <0, 3>.
Step 4: Add the motorboat's vector and the current's vector to find the resultant vector, which represents the true speed and direction of the motorboat: V_resultant = V_motorboat + V_current = <20, 0> + <0, 3> = <20, 3>.
Now we have the vector representing the speed and direction of the motorboat: <20, 3>.
To find the true speed, calculate the magnitude of the resultant vector: True speed = sqrt(20^2 + 3^2) = sqrt(400 + 9) = sqrt(409) ≈ 20.22 mph.
To find the direction, calculate the angle (θ) between the resultant vecor and the x-axis using the tangent function: tan(θ) = (3/20)
θ = arctan(3/20) ≈ 8.53 degrees.
The true speed of the motorboat is approximately 20.22 mph, and its direction is approximately 8.53 degrees from the direct path across the river.

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Answer:

151

Step-by-step explanation:

180-49(complimentary angles sum up to 180

The theoretical probability is 1/6 or approximately 0.1667. The colors are red, blue, green, yellow, purple, and orange. The theoretical probability of landing on any one of these colors with one spin is 1/6 or approximately 0.1667.

This means that if the spinner is spun many times, we would expect the spinner to land on each color approximately 1/6 of the time.  To calculate the theoretical probability, we simply divide the number of outcomes we are interested in by the total number of possible outcomes. In this case, there is one outcome we are interested in (landing on a specific color) and there are six possible outcomes (landing on any one of the six colors). Therefore,

It is important to note that theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. In reality, there may be other factors that can affect the outcomes of the spinner such as the force of the spin or the shape of the spinner. However, assuming that these factors are constant, the theoretical probability can be a useful tool in predicting the likelihood of certain events occurring.

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The proportion is solved and the equivalent value of A = 5 , when R = 3.6

Given data ,

If R is inversely proportional to A, we can write:

R = k / A

where k is a constant of proportionality. To find the value of k, we can use the given information that when A = 1.5, R = 12:

12 = k / 1.5

Multiplying both sides by 1.5, we get:

k = 18

Now we can use this value of k to find R when A = 5:

R = 18 / 5

Simplifying, we get:

R = 3.6

Hence , when A = 5, R is equal to 3.6 and the proportion is solved

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Answer:

  97 feet

Step-by-step explanation:

Given a 25-foot gorilla that Kelsey wants to stabilize with ropes at angles of 60°, 45°, and 50°, we want the total length of rope required.

When the geometry is modeled by a right triangle, the sine function relates the height of the gorilla, the rope length, and the angle by ...

  Sin = Opposite/Hypotenuse

Here, the angle is opposite the gorilla height, and the length of rope is the hypotenuse of the right triangle. For each angle, we have ...

  sin(θ) = (25 ft)/(rope length)

  rope length = (25 ft)/sin(θ)

The total rope length required will be the sum of the individual lengths for the different angles:

  total length = (25 ft)/sin(60°) +(25 ft)/sin(45°) +(25 ft)/sin(50°)

  total length = 28.9 ft +35.4 ft +32.6 ft = 96.9 ft

Kelsey needs about 97 feet of rope.

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The radius of convergence is 0, and there is no interval of convergence.

To find the radius and interval of convergence for the power series Σ(8.4 * (7 - 2)^n), n = 1 to ∞, we will use the Ratio Test. Here are the steps:

1. Write the general term of the power series: a_n = 8.4 * (7 - 2)^n.
2. Calculate the absolute value of the ratio of consecutive terms: |a_(n+1) / a_n|.
  |a_(n+1) / a_n| = |(8.4 * (7 - 2)^(n+1)) / (8.4 * (7 - 2)^n)| = |(7 - 2)^(n+1) / (7 - 2)^n|.
3. Simplify the ratio: |(7 - 2)^(n+1) / (7 - 2)^n| = |(7 - 2)| = |5|.
4. Apply the Ratio Test: For the series to converge, the ratio must be less than 1.
  |5| < 1, which is false.

Since the ratio is not less than 1, the power series does not converge for any value of x. Therefore, the radius of convergence is 0, and there is no interval of convergence.

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complete question:

Find the radius and interval of convergence for the power series 8.4" (7–2)". n=1 n.

Yes, the line should be parallel to the triangular faces.

We have,

Bella has drawn a line to represent the parallel cross-section of the triangular prism.

A cross-section is a 2-dimensional shape that is obtained by slicing a 3-dimensional object.

In the case of a triangular prism, if you slice it parallel to one of the rectangular faces, the resulting cross-section will be a rectangle.

The base of a triangular prism is a triangular face.

A "parallel cross-section" is a cross-section taken parallel to the base.

Hence, the parallel cross section should be parallel to the triangular faces.

So, the correct answer is:

Yes, the line should be parallel to the triangular faces.

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The annual tax payment for an employee at the 40% tax rate who does not make a financial contribution towards an executive car would be $20,000.

You need to know the employee's taxable income to determine the annual tax payment for a worker paying a 40% tax rate who does not contribute financially to an executive automobile.

Assume the employee does not receive any deductions or allowances, and their taxable income is $50,000 per year.

You must first determine their gross income tax, which is the amount of tax they would pay if their whole taxable income were subject to a 40% tax.

Gross Income Tax = Taxable Income x Tax Rate

Gross Income Tax = $50,000 x 0.4 = $20,000

Then, take away any deductions or allowances to which the employee is entitled. The employee's net income tax in this instance is equal to their gross income tax because it is assumed that they are not eligible for any deductions or allowances.

Therefore, the annual tax payment for an employee at the 40% tax rate who does not make a financial contribution towards an executive car would be $20,000.

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The complete question is -

What is the annual tax payment for an employee at the 40% tax rate who does not make a financial contribution towards an executive car and earns taxable income of $50,000 per year?

The value of the product [tex]$abc = 1 * 3 * 1 = 3$[/tex].

To express the number [tex]$345{,}600$[/tex] as a product of powers of prime numbers, we first need to find the prime factorization of the given number.

[tex]$345{,}600 = 2^3 * 3 * 5^3 * 23$[/tex]

Now we need to rewrite the given expression [tex]$6^a5^b4^c$[/tex] in terms of prime numbers:

[tex]$6^a5^b4^c = (2 * 3)^a * 5^b * (2^2)^c = 2^{a+2c} * 3^a * 5^b$[/tex]

By comparing the exponents of the prime factorization of [tex]$345{,}600$[/tex] with the rewritten expression, we can determine the values of [tex]$a$[/tex], [tex]$b$[/tex], and [tex]$c$[/tex]:

[tex]a+2c=3$, \\$a=1$, \\and $b=3$[/tex]

Solving for [tex]$c$[/tex], we get:

[tex]$c = (3 - a) / 2 = (3 - 1) / 2 = 1$[/tex]

Now we have [tex]a=1$, $b=3$, and $c=1$[/tex]. The product [tex]$abc = 1 * 3 * 1 = 3$[/tex].

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Answer:

adding five

Step-by-step explanation:

-8 + 5 = -3

-3 + 5 = 2

2 + 5 = 7

7 + 5 = 12

Points A and B are on side YZ of rectangle WXYZ such that WA and WB trisect ZWX. If BY = 3 and AZ = 6, then the area of rectangle WXYZ is approximately 218.23 square units.

Given that,

Points A and B are on side YZ of the rectangle WXYZ.

WA and WB trisect ZWX.

BY = 3 and AZ = 6.

From figure:

tan A = 6/y   ......(i)

And tan 2A = y/6  .....(ii)

Now tan(A) x tan(2A) = (6/y) x (y/6)

tan(A) x tan(2A) = 1

Expanding tan(2A):

[tex]tan(A) \times \dfrac{2 tan A}{1- tan^2 A} = 1\\\\3 tan^2 A = 1\\\\tanA = \dfrac{1}{\sqrt{3}} ....(iii)[/tex]

From equation (i) and (iii):

y = 6√3

Now again from the figure,

tanA = y/(x-3)

(x-3) = y/tanA

x  = 3 + 6√3√3                  from (iii)

x = 21

Therefore,

Area of the rectangle = xy,

Area of rectangle = (21)(6√3)

Area of the rectangle ≈ 218.23 square units

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According to the given triangle, the value of ∠AFE = 48 degrees. (option b).

Let's start by looking at the given figure. We have an equilateral triangle ABC, which means all its sides are equal, and all its angles are 60 degrees. We are also given a point D outside the triangle and an angle CAD of 18 degrees. This means that the angle CAD and the angle BAC are supplementary (they add up to 180 degrees), since they both share the side AC.

Now, we can apply the angle bisector theorem to the triangle AEB. This tells us that AE/EB = AD/DB. Since CD = BE and triangle ADE is congruent to triangle BDC, we can say that

AD/DB = AE/EC = (AE + EC)/EC = AC/EC.

Therefore,

=> AE/EC = AC/EC - 1 = (2 cos 18 - 1)/sin 18.

Solving for sin ∠AFE, we get sin

∠AFE = (sin 12)(sin (42 - x))/sin (126 - x)(sin (60 - x)).

Taking the inverse sine of this value, we get

∠AFE = 48 degrees.

Therefore, the answer is option B, 48 degrees.

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Weight of an apple: quantitative continuous.The most common color of apples in a bag: qualitative.Number of apples in a two-pound bag: quantitative discrete.

1. Weight of an apple: The weight is a measurable quantity with numerical values. Therefore, this data is quantitative. Since weight can take on any value within a range (e.g., 5.2 oz, 5.25 oz), it is continuous. So, the data is quantitative continuous.

2. The most common color of apples in a bag: Color is a non-numerical characteristic that describes the apples. This data is qualitative.

3. Number of apples in a two-pound bag: The number of apples is a countable numerical value. This data is quantitative. Since it can only be a whole number (e.g., 5 apples, 6 apples), it is discrete. So, the data is quantitative discrete.

Your answer:

1. Weight of an apple - Quantitative continuous
2. The most common color of apples in a bag - Qualitative
3. Number of apples in a two-pound bag - Quantitative discrete

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Answer:

n=1\2

Step-by-step explanation: