Alex and Tory are married and filing jointly. Their taxable income is $150,000. Based
on the table below, how much do they owe in federal taxes?
Note: For a married couple filing jointly, the standard deduction is $25,900.

The dependent and independent variables are b and t respectively.

A variable is a mathematical symbol, which do not have any fixed value, it can be a function, which changed according to the property given.

Given that, a committee is organizing a music festival in Jefferson County,

The time which reserved by venue, gives the number of band that will play at there,

We need to determine the dependent and independent variables,

Dependent variable :-

Dependent variable is a kind of variable which depends on the factors given, and do not change by its own.

Independent variables :-

The independent variable in the given study is the cause by which the dependent variable works, it does not manipulate by any other variable given.

Here,

The number of bands depends upon the amount of the time that the venue has been reserved for.

Therefore, the number of band is a dependent variable.

Hence, the dependent and independent variables are b and t respectively.

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a).north component = 44.0 km/h * sin(40.0°) ≈ 28.31 km/h

East component = 44.0 km/h * cos(40.0°) ≈ 33.71 km/h

The difference between the northern components of the fishing boat and the speedboat is:

North difference = 28.31 km/h - 0 km/h = 28.31 km/h

So the time it takes for the speedboat to intercept the fishing boat is:

Time = North distance / North difference = 6.21 km / (28.31 km/h) = 0.219 hours

Converting to minutes:

Time = 0.219 hours * 60 minutes/hour ≈ 13.1 minutes

Therefore, it takes the speedboat about 13.1 minutes to intercept the fishing boat.

b) tan θ = east component / north component

θ = tan⁻¹(east component / north component)

θ ≈ 51.3°

So the direction in which the speedboat travels is 51.3° east of north. Therefore, the compass bearing with respect to due north is:

Bearing = 90° - 51.3° ≈ 38.7° east of north.

In mathematics, a function is a relationship between two sets of values, such that each input value (also known as the argument or independent variable) is associated with exactly one output value (also known as the value or dependent variable). A function is usually denoted by a symbol such as "f(x)" or "y" and is defined by a rule or formula that specifies how the input value is transformed into an output value.

For example, consider the function f(x) = 2x. This function takes an input value x, multiplies it by 2, and returns the result as the output value. So, for example, when x is 3, the output value is 6. When x is 5, the output value is 10.

Functions can be represented graphically as well. The graph of a function is a set of points in a two-dimensional coordinate system, where the x-coordinate is the input value and the y-coordinate is the output value. For example, the graph of the function f(x) = x^2 is a parabola.

Functions are widely used in many areas of mathematics, science, engineering, economics, and more. They provide a powerful tool for modeling real-world situations, making predictions, and analyzing data.

(a) To find how much time it takes for the speedboat to intercept the fishing boat, we first need to find the position of the fishing boat at the moment the speedboat launches.

From the lighthouse's perspective, the fishing boat is located at a bearing of 15.0° east of north and a distance of 24.0 km. Using trigonometry, we can find the north and east components of the fishing boat's position:

North component = 24.0 km * sin(15.0°) ≈ 6.21 km

East component = 24.0 km * cos(15.0°) ≈ 22.76 km

Now we can use the relative velocity between the fishing boat and the speedboat to find the time it takes for the speedboat to intercept the fishing boat. The speedboat's velocity can be broken down into north and east components:

North component = 44.0 km/h * sin(40.0°) ≈ 28.31 km/h

East component = 44.0 km/h * cos(40.0°) ≈ 33.71 km/h

The difference between the northern components of the fishing boat and the speedboat is:

North difference = 28.31 km/h - 0 km/h = 28.31 km/h

So the time it takes for the speedboat to intercept the fishing boat is:

Time = North distance / North difference = 6.21 km / (28.31 km/h) = 0.219 hours

Converting to minutes:

Time = 0.219 hours * 60 minutes/hour ≈ 13.1 minutes

Therefore, it takes the speedboat about 13.1 minutes to intercept the fishing boat.

(b) To find the direction in which the speedboat travels, we can use trigonometry to find the angle between the speedboat's velocity vector and the north direction.

The north component of the speedboat's velocity is 28.31 km/h, and the east component is 33.71 km/h. Using the tangent function, we can find the angle:

tan θ = east component / north component

θ = tan⁻¹(east component / north component)

θ ≈ 51.3°

So the direction in which the speedboat travels is 51.3° east of north. Therefore, the compass bearing with respect to due north is:

Bearing = 90° - 51.3° ≈ 38.7° east of north.

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The people may easily accept the conclusion of the study described because of a common misunderstanding of correlational studies.

The Correlational Studies are used to examine the relationship between two variables, and a positive correlation suggests that the two variables are related in a certain way .

In this study, the Researchers found a strong positive correlation between sugary snacks and hyperactivity, suggesting that as the amount of sugary snacks offered increased, the number of times children left their seats also increased.

Therefore , people easily accept the conclusion of the study that sugar causes hyperactivity due to a misunderstanding of correlational studies and a lack of knowledge about the need for experimental designs to establish causality.

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The given question is incomplete , the complete question is

Researchers conducted a naturalistic study of children between the ages of 5 and 7 years. The researchers visited classrooms during class party celebrations. As a measure of hyperactivity, they recorded the number of times children left their seats. The researchers found a strong positive correlation between sugary snacks offered at the parties and hyperactivity.

Based on these findings, the researchers concluded that sugar causes hyperactivity.

Explain why people may easily accept the conclusion of the study described above? Include in your explanation a misunderstanding of correlational studies

Vogel's approximation method is used to find an initial feasible solution to a transportation problem.

The method is based on the observation that if there is a row or a column in the transportation tableau having the same minimum positive difference (or penalty) between the costs of two cells, it is likely that one of these cells will get a positive allocation in the optimal solution.

Calculate the penalty for each cell by subtracting the smaller of the two adjacent cells from the larger.

D1 D2 D3 D4 A B C D

5 1 3 3 34 3 6 4

Penalty 0 2 0 1 31 2 2 -1

3 3 5 4 15 3 4 -1

Penalty 0 0 2 1 12 2 2 2

6 4 4 3 12 4 3 2

Penalty 2 1 1 0 6 1 0 1

4 -1 4 2 19 2 3 2

Penalty 5 5 2 0 17 2 0 0

Identify the row and column having the maximum penalty. In this case, the maximum penalty is 5 in row 1 and column 4.

Allocate as much as possible to the cell in the intersection of the row and column with the maximum penalty. The allocation should not exceed the demand or the supply of that row or column.

D1 D2 D3 D4 A B C D

5 1 3 2 34 3 6 4

Penalty 0 2 0 1 31 2 2 -1

3 3 5 3 15 3 4 -1

Penalty 0 0 2 1 12 2 2 2

6 4 4 3 12 4 3 2

Penalty 2 1 1 0 6 1 0 1

4 -1 4 2 19 2 3 2

Penalty 5 5 2 0 17 2 0 0

Repeat the process until all demands are met or all supplies are exhausted.

D1 D2 D3 D4 A B C D

5 1 2 2 34 3 6 3

Penalty 0 2 0 0 32 2 2 -1

3 3 5 3 15 3 3 0

Penalty 0 0 2 0 12 2 1 2

6 4 4 3 12 4 0 1

Penalty 2 1 1 0 6 1 0 0

4 -1 4 2 19 2 3 1

Penalty 5 5 2 0 17 2 0 0

The above table shows an initial feasible solution to the transportation problem using Vogel's approximation method. The total cost of this solution is 34 + 15 + 12 + 19 = 80, which is the same

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Answer:

Step-by-step explanation:

To find the average speed for the round trip, we can use the formula:

average speed = total distance / total time

We know that Steve drives 5 miles to school and 5 miles back home, so the total distance is:

total distance = 5 miles + 5 miles = 10 miles

To find the total time, we need to calculate the time it takes for Steve to drive to school and the time it takes for him to return home. We can use the formula:

time = distance / speed

For the first part of the trip, Steve drives 5 miles at 10 mph, so the time it takes is:

time to school = 5 miles / 10 mph = 0.5 hours

For the second part of the trip, Steve drives 5 miles at 40 mph, so the time it takes is:

time to home = 5 miles / 40 mph = 0.125 hours

The total time for the round trip is the sum of the time to school and the time to home:

total time = time to school + time to home

total time = 0.5 hours + 0.125 hours

total time = 0.625 hours

Now we can calculate the average speed using the formula:

average speed = total distance / total time

average speed = 10 miles / 0.625 hours

average speed = 16 miles per hour (rounded to the nearest integer)

Therefore, Steve's average speed for the round trip is 16 mph.

Answer:

16 mph

Step-by-step explanation:

The probability of a positive test result given that the subject actually lied is 0.962

In simple terms, probability is the measure of the likelihood of an event occurring. In this question, we are asked to find the probability of a positive test result given that the subject lied.

Let's break down the question and use the formula for conditional probability. Conditional probability is the probability of an event occurring, given that another event has already occurred.

P(A|B) = P(A and B) / P(B)

In this case, A represents the event of a positive test result, and B represents the event of the subject lying.

From the question, we know that 2% of the subjects lie, and 90% of those who lie test positive. This means that out of the 98 subjects, 2% or 1.96 subjects lied. And out of those 1.97 subjects, 90% or 1.78 subjects tested positive.

So, the probability of a positive test result given that the subject lied is

=> P(A|B) = 1.78/1.97 = 0.962.

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Complete Question:

If one of the 98 test subjects is randomly selected, find the probability that the subject had a positive test result GIVEN

POSITIVE TEST RESULT      13            40

NEGATIVE TEST RESULT    34            11

that the subject actually lied.

A 0.962

B 0.654

C 0.784

D 0.456

The missing side length of the figure is given as follows:

6 cm.

The segment addition postulate is a geometry axiom that states that a segment, divided into a number of smaller segments, has the length given by the sum of the lengths of the segments.

For this problem, we have that:

Hence the missing side length of the figure is obtained as follows:

16 - 10 = 6 cm.

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Reason: It fails the vertical line test. It is possible to pass a single vertical line through multiple points on this blue graph.

For instance, we can have a vertical line through x = 1. This vertical line intersects infinitely many points.

In other words, the input x = 1 leads to more than one output, which is a counter-example to show we do not have a function.

A function is only possible if each input in the domain leads to exactly one output in the range.

Since we don't have a function, you don't need to worry about filling in the domain and range boxes.

The graph is a function with a domain of (-infinity, infinity) and a range of (-infinity, infinity).

The graph represents a function because for each value of x, there is exactly one corresponding value of y. As a result, the vertical line test is passed.

The domain of the function is the set of all x-values for which the function is defined. In this case, it appears that the graph extends from x = -infinity to x = infinity, so the domain is (-infinity, infinity).

The function's range is the set of all y-values that the function can yield. According to the graph, the y-values vary from y = -infinity to y = infinite, resulting in the range (-infinity, infinity).

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Answer:

Since triangle ABC is similar to triangle FGH, their angle measures are proportional. We can use the ratios of their angle measures to find the missing angle measures.

Let's call the missing angle measures x, y, z, and w, where x is m∠F, y is m∠G, z is m∠H, and w is m∠C.

From the given information, we have:

m∠A / m∠F = 22 / x

m∠B / m∠G = 75 / y

Since m∠A + m∠B + m∠C = 180 degrees, we can write:

w = 180 - (m∠A + m∠B)

m∠H / m∠C = z / w

Now we can use the ratios to find the missing angle measures:

x = 22 / (22 / x) = 22

y = 75 / (75 / y) = 75

z = m∠H / (m∠H / z) = m∠H

w = 180 - (22 + 75) = 83

So, the missing angle measures are:

m∠F = 22 degrees

m∠G = 75 degrees

m∠H = m∠H (unknown)

m∠C = 83 degrees

The correct responses are:

If you have SSA, the triangle is determined by the third side. It does not form a triangle if it is too short to intersect the other side. A triangle can also be formed if the third side is the ideal length to connect to the other two sides. Last but not least, the third side may be long enough to intersect the opposite side twice, forming two triangles.

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complete question:

When using the law of sines, why can the SSA case result in zero, one, or two triangles?

All the areas of the circles are illustrated below.

The circumference of a circle is the distance around the circle which is 2πr.

The diameter of a circle is the largest chord that passes through the center of a circle it is 2r.

We know, The area of the circle is πr².

1. The area of the circle with a radius of 2.5 cm is,

= π(2.5)² sq cm.

= 19.625 sq cm.

2. The area of the circle with a radius of 11 in is,

= π(11)² sq in.

= 379.94 sq in.

3. The area of the circle with a radius of 3 mm is,

= π(3)² sq mm.

= 28.26 sq mm.

4. The area of the circle with a radius of 5 in is,

=  π(5)² sq in.

= 78.5 sq in.

5.  The area of the circle with a radius of 6.5 cm is,

= π(6.5)² sq cm.

= 132.665 sq cm.

6.   The area of the circle with a radius of 7.2 yd is,

= π(7.2)² sq yd.

= 162.7776 sq yd.

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The rate of change on a slope graph is 25.

How do you find the rate of change on a slope graph?

The rate of change for a line is the slope, the rise over run, or the change in over the change in. The slope can be calculated from two points in a table or from the slope triangle in a graph.

The average rate of change formula is used to find the slope of a graphed function. To find the average rate of change, divide the change in y-values by the change in x-values.

Given :

Y - variations = 0 , 25 , 50 ,75 , 100 , 125..........

X - variations = 0 , 1 , 2 , 3 , 4 , 5 , 6 .........

Thus , we can formulate the points as,

(0,0) , (1,25) , (2,50) , (3,75) and so on ...

for rate of change of graph with (x1 , y1) and (x2 , y2) as co-ordinate point we know that,

rate of change = (y2-y1)/(x2-x1)

Similarly ,by taking any two points(let (0,0) and (1,25)) from formulated points , we can say that

rate of change = (25-0)/(1-0)

rate of change = 25

Hence , the rate of change on a slope graph is 25.

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135 in Mayan notation is represented by one dot over four bars, followed by one dot over three bars, and then one dot over five dots.

To convert 135 to Mayan notation, we repeatedly divide by 20 and use the remainders to determine the number of dots and bars in each position. First, we divide 135 by 20 to get a quotient of 6 with a remainder of 15. The remainder of 15 corresponds to 1 dot over 5 dots and 2 bars (10 + 5).

Next, we divide 6 by 20 to get a quotient of 0 with a remainder of 6. The remainder of 6 corresponds to 1 dot over 3 bars (15). Finally, we have a quotient of 0 with a remainder of 6, which corresponds to 1 dot over 4 bars (20).

Putting these together, we get the Mayan representation of 135 as one dot over four bars, followed by one dot over three bars, and then one dot over five dots.

Complete question:

Convert the following numbers to Mayan notation. Show your calculations used to get your answers. 135?

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The ratio of games won to games played is options A and E, 16/20 or 4/5 in fraction form and 0.8 in decimal form.

Ratio is defined as the relationship between two quantities where it tells how much one quantity is contained in the other.

The ratio of a and b is denoted as a : b.

Given that,

For every 16 games the football team won, they lost four.

Games won = 16

Games lost = 4

Total games played = 16 + 4 = 20

Ratio of games won to total games in fractional form = 16 / 20 = 4/5

Ratio of games won to total games in decimal form = 0.8

The correct options are A and E.

Hence the ratio of games won to total games is 16/20 or 4/5 in fraction form and 0.8 in decimal form.

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Your question is incomplete. The complete question is given below.

Manchester Football team participates in a tournament. For every 16 games the football team won, they lost four. Represent the ratio of games won to games played in fraction form and decimal form.

Select TWO correct answers.

A. 4/5, 0.8

B.  1/2, 0.5

C.  2/4, 0.5

D. 1 /4, 0.25

E.  16/20, 0.8

Answer:

3000 liters

Step-by-step explanation:

To calculate the volume of the tin can, we can multiply its dimensions:

Volume = 2m × 1.5m × 1m = 3 cubic meters

Since a liter is equivalent to 0.001 cubic meters, we can convert the volume of the tin can to liters:

Volume (in liters) = 3 cubic meters × 1000 liters/cubic meter = 3000 liters

Therefore, the tin can can hold 3000 liters of kerosene oil.

Aɳʂɯҽɾҽԃ Ⴆყ ɠσԃKEY ꦿ

The values of data measured on this scale are labels or names. -> Nominal Scale

The values of data measured on this scale can be rank ordered. -> Ordinal Scale. The values of data measured on this scale can be rank ordered and have meaningful an absolute zero point. -> Ratio Scale

The values of data measured on this scale can be rank ordered. In addition, the differences between two adjacent ranks are equal. -> Interval Scale

Nominal Scale: In the nominal scale, the values of data are labels or names used to categorize or classify items into mutually exclusive groups. Nominal data cannot be ordered, measured, or compared. Examples of nominal data include gender (male, female), eye color (blue, brown, green), or car brands (Toyota, Honda, Ford).

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The 22 half-pieces Kristin had from cutting the sheets she started with and the 7 pieces she found, indicates, that the solution to the word problem is that the number of sheets of paper Kristin started with are;

A word problem is a mathematical question in which the scenario of the question is described using verbal terms, or complete sentences, rather than mathematical symbols or expressions, but which are solved using mathematical calculations.

Let x represent the initial number of paper Kristin had.

The extra number of papers Kristin found = Seven more pieces

The size in which Kristin cut each piece of paper = In half

The number of pieces she had after cutting the papers = 22 half-pieces

Therefore, the following equation can be used to find the number of sheets of paper, x, she started with;

2 × (x + 7) = 22

2 × (x + 7)/2 = 22 ÷ 2 = 11

x + 7 = 11

x = 11 - 7 = 4

The number of sheets of paper Kristin started with, x = 4 sheets of paper

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The angles are vertically opposite angles and the measure of x = 81

Two straight lines that intersect at the same location are said to be intersecting lines. The junction point is the place where two intersecting lines meet. Four angles are created when two lines cross. The four angles added together always equal 360 degrees.

Perpendicular lines are two straight lines that intersect and form right angles. When two perpendicular lines intersect, they form four right angles.

When lines intersect, two angle relationships are formed:

Opposite angles are congruent

Adjacent angles are supplementary

Given data ,

Let the first line be represented as m

Let the second line be represented as n

Now , the lines m and n intersect at point O and the resulting lines are formed

where ∠A = ∠B ( vertically opposite angles )

So , the measure of ( x + 3 )° = 84°

On simplifying , we get

x + 3 = 84

Subtracting 3 on both sides , we get

x = 81

Hence , the measure of angles are opposite angles

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a) The probability of scoring a one is 1/7, since a one is not the loaded number six and all other scores are equally likely.

b) The probability of scoring a six is 3/7, since a six is the loaded number and occurs three times as often as any other score.

Let p be the probability of scoring any number except six. Then the probability of scoring a six is 3p, since a six is scored three times more often than any other score. Since there are six equally likely possible scores on a die, we have:

p + 3p + p + p + p + p = 1

Simplifying, we get:

7p = 1

Dividing both sides by 7, we get:

p = 1/7

Therefore, the probability of scoring any number except six is 1/7, and the probability of scoring a six is 3/7. We can now answer the questions:

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The Exact values for the sine and cosine of the given angles are : sin(120°) = √3/2, cos(120°) = -1/2

Given that the terminal ray of an angle with measure of 120 degrees intersects a unit circle at the point (-1/2, √3/2), we can use the definition of sine and cosine to find the exact values of these trigonometric functions for the given angle.

The sine of an angle is defined as the y-coordinate of the point on the unit circle that the terminal ray of that angle intersects. So, for this angle, the sine is:

sin(120°) = √3/2

The cosine of an angle is defined as the x-coordinate of the point on the unit circle that the terminal ray of that angle intersects. So, for this angle, the cosine is:

cos(120°) = -1/2

So, the exact values for the sine and cosine of the angle with measure of 120 degrees are:

sin(120°) = √3/2

cos(120°) = -1/2

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The area of the circle in discuss as required to be determined is; 44/7.

It follows that the radius of a circle is the distance between its center and any point on the circumference.

In this case, center, = (0, 0) and point in circumference= (-1, -1).

Therefore, the radius is;

r = √( (-1 -0)² + (-1-0)² )

r = √2.

Consequently, since area of a circle is given by;

Area = πr²

A = (22/7) × (√2)²

A = 44/7.

Ultimately, the area of the circle is; 44 / 7.

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Step-by-step explanation: A stem and leaf plot is a way to plot data where the data is split into stems (the largest digit) and leaves (the smallest digits). They were widely used before the advent of the personal computer, as they were a fast way to sketch data distributions by hand.

Answer:

Part A: length = 4x - 5

Part B: See explanation below

Step-by-step explanation:

Part A

Area of a rectangle = length x width

Given area and width we can find the length as

[tex]length = \dfrac{area}{width}[/tex]

[tex]Area = 12x^2 - 15x\\\\Width = 3x\\\\Length = \dfrac{12x^2 - 15x}{3x}\\\\= \dfrac{12x^2}{3x} - \dfrac{15x}{3x}\\\\= 4x - 5\\\\[/tex]

Answer to Part A

Part B

[tex]length = 4x- 5\; (from part A)}[/tex]

[tex]width = 3x \;(given)}[/tex]

[tex]area = length \times width[/tex]

[tex]=(4x - 5)(3x)\\\\= 4x(3x) - 5(3x)\\\\= 12x^2 - 15x\\\\[/tex]

Hence verified

Answer:

To construct a 90% confidence interval for the mean amount of juice in all such bottles, we first need to find the sample mean and sample standard deviation:

Sample mean, x = (15.2 + 15.5 + 15.9 + 15.5 + 15.0 + 15.7 + 15.0 + 15.7)/8 = 15.4375

Sample standard deviation, s = s = sqrt[((15.2-15.4375)^2 + (15.5-15.4375)^2 + (15.9-15.4375)^2 + (15.5-15.4375)^2 + (15.0-15.4375)^2 + (15.7-15.4375)^2 + (15.0-15.4375)^2 + (15.7-15.4375)^2)/7] = 0.339

Using a t-distribution with degrees of freedom (n-1) = 7 and a 90% confidence level, we can find the t-value as 1.895.

The 90% confidence interval can then be calculated as:

x plus or minus (t-value)*(s/sqrt(n))

= 15.4375 plus or minus (1.895)*(0.339/sqrt(8))

= (15.16, 15.72)

Therefore, the answer is A.

To find a 99% confidence interval for the mean monthly income for all workers at the plant, we use the formula:

x plus or minus (z-value)*(σ/sqrt(n))

where x is the sample mean, σ is the population standard deviation, n is the sample size, and z-value is the critical value from the standard normal distribution for a 99% confidence level, which is 2.576.

Plugging in the given values, we get:

x  plus or minus (z-value)*(σ/sqrt(n))

= 2150  plus or minus (2.576)*(250/sqrt(18))

= (2096, 2204)

Therefore, the answer is A.

Every point of an open set in R2 is a limit point, but only the boundary points of a closed set are limit points.

Every point of every open set E in R2 is a limit point of E. This is because an open set is characterized by the fact that all of its boundary points are included in the set. Therefore, all of the points in the set are limit points because they are all on the boundary of the set.

For closed sets in R2, the answer is not the same. A closed set is characterized by the fact that it is determined by its boundary points and all of its interior points. Therefore, only the boundary points of a closed set are limit points, while the interior points are not.

Every point of an open set in R2 is a limit point, but only the boundary points of a closed set are limit points.

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To determine which expression has a value of 18, we can evaluate each expression and see which one equals 18.

5 × (40 – 25) – 50 = 5 × 15 – 50 = 75 – 50 = 25, which is not equal to 18.

6x (3212) (100 +50) - (6 + 18+6) + 15 + 10 = 6 × 3212 × 150 - 30 + 25 = 29,021,770, which is not equal to 18.

07+ (16-7)+3+8 = 7 + 9 + 3 + 8 = 27, which is not equal to 18.

Therefore, none of the given expressions has a value of 18.

Based on the given option, the correct answer would be; C. 2x - 3y = 6 and 2x + y = -6

An equation is an expression that shows the relationship between two or more numbers and variables.

A mathematical equation is a statement with two equal sides and an equal sign in between. An equation is, for instance, 4 + 6 = 10. Both 4 + 6 and 10 can be seen on the left and right sides of the equal sign, respectively.

here, we have,

We are given the system of equations as;

2/3x - y = 2

x + 1/2 y = -3

Here multiply by 3 on both sides;

2x - 3y = 6

Now similarly;

x + 1/2 y = -3

2x + y = -6

The result would be C. 2x - 3y = 6 and 2x + y = -6

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Answer:

Since we are given the standard deviation of the sample, we can use the formula:

variance = standard deviation^2

Substituting the given value:

variance = 10.3695^2 kg^2

Calculating:

variance ≈ 107.495 kg^2

Therefore, the variance of the weights of the 10 subjects is approximately 107.495 kg^2.

The probability density function for the average length of life of the two components is fU(u) = (1/7)[tex]e^{(-u/7)}[/tex], u ≥ 0

The length of life of each component is governed by an exponential distribution with a mean of 7, which means that the probability density function for the length of life of each component is given by:

fX(x) = (1/7)[tex]e^{(-x/7)}[/tex], x ≥ 0

The average length of life of the two components is given by:

U = (X1 + X2)/2

where X1 and X2 are the lengths of life of the two components.

To find the probability density function for U, we can use the convolution formula:

fU(u) = ∫fX(x)fX(2u-x)dx

where the limits of integration are from 0 to u if u ≤ 7, and from u-7 to 7 if u > 7.

Plugging in the expressions for fX(x) and fX(2u-x), we get:

fU(u) = ∫(1/7)[tex]e^{(-x/7)}[/tex](1/7)[tex]e^{-(2u-x/7)}[/tex]dx

= (1/49)∫[tex]e^{-(2u-2x/7)}[/tex]dx

= (1/49)∫[tex]e^{(-t/7)}[/tex]dt (where t = 2u - 2x)

= (1/49)(-7[tex]e^{(-x/7)}[/tex])|0 to 2u

= (1/7)[tex]e^{(-u/7)}[/tex], u ≥ 0

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