A cone has base area 25 mm². A parallel slice 6 mm from the vertex has area 36 mm². Find the height of
the cone.

In response to the query, we can state that Hence, P"s coordinates are coordinates  (3, 4).

When locating points or other geometrical objects precisely on a manifold, such as Euclidean space, a coordinate system is a technique that uses one or more integers or coordinates. Locating a point or item on a two-dimensional plane requires the use of coordinates, which are pairs of integers. Two numbers called the x and y coordinates are used to describe a point's location on a 2D plane. a collection of integers that represent specific locations. The figure often has two numbers. The first number denotes the front-to-back measurement, while the second number denotes the top-to-bottom measurement. For example, in (12.5), there are 12 units below and 5 above.

A point's y-coordinate changes sign when it is reflected across the x-axis, but its x-coordinate stays the same.

Hence, we must modify the sign of the y-coordinate while leaving the x-coordinate unaffected in order to reflect point P(3,-4) across the x-axis. The reflected point P' will therefore have coordinates (3, 4).

Hence, P"s coordinates are (3, 4).

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The area of the entire rectangle is 20x^2 - 10x - 30

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

Height = 5

Width = 4x^2 - 2x - 6

Using the above as a guide, we have the following:

The area of a rectangle is the product of its dimension

So, we have

Area = Height * Width

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

Area = 5 * (4x^2 - 2x - 6)

Evaluate

Area = 20x^2 - 10x - 30

Hence, the area is 20x^2 - 10x - 30

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Hence, in response to the provided question, we can say that As a result,  arithmetic progression the usual difference is d = -30.

The difference between subsequent terms in a string will never vary, and this is referred like an arithmetic progression. The sequences 5, 7, 9, 11, 13, through 15 are examples of arithmetic progressions with a tolerance of 2. Arithmetic progression (A.P.) is a progression that has a set tolerance between any 2 consecutive values. Numerical progression may fall into one of two main types: finite-length arithmetic series A finite geometry progression is a series with just a fixed number of terms. The series' terms can be used to calculate the early, late, leniency, and number of term.

A) The sequence is, indeed, arithmetic.

We can find the common difference by subtracting any two consecutive terms:

109 - 9 = 100

209 - 109 = 100

309 - 209 = 100

As a result, the typical difference is d = 100.

A) The sequence is, indeed, arithmetic.

We can find the common difference by subtracting any two consecutive terms:

-32 - (-2) = -30 \s-62 - (-32) = -30 \s-92 - (-62) = -30

As a result, the usual difference is d = -30.

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Volume of Sphere B is 8 times greater than the volume of Sphere A

What does sphere refer to?

A sphere is symmetrical and round in form. All of its surface points are equally spaced from the centre, making it a three-dimensional solid. Depending on the radius, it has a surface area and a volume. No faces, corners, or edges exist on it.

                            It is sometimes referred to as the second cousin of a circle, the sphere is a three-dimensional form. Round, without boundaries, and solid in shape, a sphere is spherical. Examples of spherical shapes include a playing ball, a balloon, and even light bulbs.

Sphere A:  V = 4π (2³)/3 = 32π/3

Sphere B:  V = 4π (4³)/3 = 4π(64)/3 = 256π/3

volume of Sphere B / volume of Sphere A =  (256π/3)/(32π/3) = 8

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The data collected by the ministry of health denotes C) Observational Study

The degree of movement along or opposite that may occur in one of the data values when another data value is increased or lowered, respectively, is shown by the correlation of two pairs of data values. An observational study selects participants from a sample of the community, but the independent variable is not controlled due to moral considerations, outside influence, and occasionally, restrictions.

To gather data on office employment and worker health concerns, the health ministry only provides questionnaires to employees every five years. They are only keeping an eye on the employees, and the survey enables them to evaluate and draw a connection between working in an office and health issues among employees. This makes the study an observational one.

Complete Question:

The ministry of health was interested in the relationship between the office habits of workers and health issues. Each year, they distributed a survey to all workers in the public sector that contained various questions on work habits as well as a standard health questionnaire. After 5 years they gathered and analyzed the entire data.

A) Sample study

B)Experiment

C) Observational Study

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Answer:

angle=80 degrees

Step-by-step explanation:

if we know the arc is 160 degrees than we know we want to find the interior angle, then we must divide 160/2 to get the measure of the indicated angle.

If the company has limited resources or is not particularly concerned with precision, it may decide to forego the additional information and rely on its prior probabilities to make decisions

Bayes TheoremBayes Theorem is a statistical technique that enables one to find the probability of an event given another event that is known to have happened. The theorem states that the probability of an event A happening given that another event B has already occurred is equivalent to the probability of event B happening given that event A has already occurred multiplied by the probability of A happening divided by the probability of B happening. The revised decision tree can be produced to make a decision. However, before that, we need to find the sum of getting a Raise report for each outcome of the prior probabilities. This is shown in the table below:Consultant`s Prediction (in %)Actual Market OutcomePrior ProbabilitiesRiseStableFallRaise0.60.70.20.10.70.140.1260.0280.0140.20.30.20.60.240.0600.0180.1260.03 0.03150.0060.00210.07Total 0.34 0.36 0.30 1.00To revise the consultant's prediction using Bayes theorem, we can use the following formula:P(Stable | Raise) = (P(Raise | Stable) * P(Stable)) / (P(Raise | Stable) * P(Stable) + P(Raise | Rise) * P(Rise) + P(Raise | Fall) * P(Fall)) = (0.140 * 0.30) / (0.140 * 0.30 + 0.126 * 0.34 + 0.028 * 0.36) = 0.2906 or 29.06%The probability of getting a stable report given a raise report is 29.06%. This value can also be calculated for getting a fall report given a raise report using the same formula, which gives a probability of 24.51%.To make a decision on whether to use the consultant firm or not, we can use the revised decision tree. The tree shows that the probability of getting a raise report is only 34%, and given that we get a raise report, the probability of getting a stable report is 29.06% while the probability of getting a fall report is 24.51%.If the company decides to use the consultant firm, they will have to pay for the additional information. This cost needs to be weighed against the benefits of having this information, which will depend on the goals and priorities of the company. If the company values precision and accuracy in its decision-making process and is willing to pay for it, then it may choose to use the consultant firm. However, if the company has limited resources or is not particularly concerned with precision, it may decide to forego the additional information and rely on its prior probabilities to make decisions.

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Answer:

see below

Step-by-step explanation:

350=[tex]\frac{4}{3}[/tex][tex]\pi[/tex]r[tex]2^{2}[/tex]

^ is this correct. i will edit my answer to be the solution.

Answer:

[tex]32x-16[/tex]

Step-by-step explanation:

[tex](2x+3)^{2} - (2x-5)^{2}[/tex]

expand the equation.

[tex](2x+3)(2x+3) - (2x-5)(2x-5)[/tex]
then distribute.
[tex](4x^{2} +6x + 6x+9) - (4x^2 - 10x - 10x + 25)[/tex]

finally simplify the equation.

= [tex]32x-16[/tex]

Answer: B - A kitten who is just born is predicted to weigh 4.05 ounces

Step-by-step explanation:

The change in the amount of shore after 7 years and 4 months is -5.4975 feet.

First, let us convert the values into the same quantities

7 1/3 years = 7 + 1/3 years = 7.33 years

The change in shore after 7 years and 4 months can be calculated as

Change in amount of shore = (average change per year) x (number of years)

Change in amount of shore = (-3/4 foot/year) x (7.33 years)

Change in amount of shore = -5.4975 feet.

Therefore, the change in the amount of shore after 7 years and 4 months is approximately -5.4975 feet.

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A system of inequalities to represent the constraints of this problem are x ≥ 0 and y ≥ 0.

A graph of the system of inequalities is shown on the coordinate plane below.

In order to write a system of linear inequalities to describe this situation, we would assign variables to the number of HD Big View television produced in one day and number of Mega Tele box television produced in one day respectively, and then translate the word problem into algebraic equation as follows:

Since the HD Big View television takes 2 person-hours to make and the Mega TeleBox television takes 3 person-hours to make, a linear equation to describe this situation is given by:

2x + 3y = 192.

Additionally, TVs4U’s total manufacturing capacity is 72 televisions per day;

x + y = 72

For the constraints, we have the following system of linear inequalities:

x ≥ 0.

y ≥ 0.

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The statement "The basic calculation for obtaining a response bias is not to divide the number of legitimately completed questionnaires returned by the total number of sample members provided a chance to participate." is False

Instead, the basic calculation for obtaining a response bias is to subtract the response rate from 100%. The response rate is calculated by dividing the number of legitimately completed questionnaires returned by the total number of sample members provided a chance to participate.

So, the formula for calculating response bias is:
Response bias = 100% - (number of legitimately completed questionnaires returned / total number of sample members provided a chance to participate)

In other words, response bias is the percentage of sample members who did not participate or did not complete the questionnaire legitimately. It is important to calculate response bias in order to understand how representative the sample is of the population.

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The spread of Carter's social media post is f(x) = 3(3)x. On Day 3, Ben's post will be received 8 shares and Carter's post will be received 27 shares. On Day 10, Ben's post will be received 524 shares and Carter's post will be received 729 shares.

Amber's new function of f(x) = 3(4)x + 45 will produce a graph with a steeper slope than the original graph of Amber's photo share.  Carter's post travels the fastest, as his equation form has the highest exponential rate. Depending on the goal of the post, one might prefer a post with fewer friends initially but more shares, such as Amber, or more friends initially but fewer shares, such as Ben.

Exponential function is a mathematical function that increases or decreases exponentially. It is an equation of the form y = ax^b, where a is the base and b is the exponent, and x is the independent variable.

1. The exponential function to represent the spread of Ben’s social media post is f(x) = 3(2)x.

2. The exponential function to represent the spread of Carter’s social media post is f(x) = 3(3)x.

3. The graph of Ben’s post will have a curve that increases exponentially as the x-values increase, while the graph of Carter’s post will have a curve that increases exponentially faster than Ben’s as the x-values increase. Three points that can be used to graph each curve are (0, 3), (1, 9), and (2, 27) for Ben’s post and (0, 3), (1, 9), and (2, 27) for Carter’s post.

4. On Day 3, Ben’s post will have 81 shares (f(3) = 3(2^3) = 81), and Carter’s post will have 243 shares (f(3) = 3(3^3) = 243). On Day 10, Ben’s post will have 59049 shares (f(10) = 3(2^10) = 59049), and Carter’s post will have 59049 shares (f(10) = 3(3^10) = 59049). This is shown in the equation form of the functions because each function is in the form of a(b^x). For Ben’s post, a = 3, b = 2, and for Carter’s post, a = 3, b = 3. As the x-values increase, the value of b increases exponentially and so the total number of shares increases exponentially.

5. Amber’s new function representing her photo shares is f(x) = 3(4)x + 45. The graph of this function will be different from the original graph of Amber’s photo share because it is no longer an exponential function - it is a linear function. The graph will have a constant rate of increase as the x-values increase, reflecting that the number of shares is increasing by a constant number with each additional day.

6. Based on the results, Carter’s post travels the fastest. This is shown in the equation form of the functions because Carter’s post has a larger value of b (3) than Ben’s post (2), indicating that Carter’s post is increasing exponentially faster than Ben’s post.

7. It would depend on the situation, but generally speaking it would be preferable to have more friends initially and fewer shares, like Amber. This is because if a post has few friends initially and more shares, the post will spread faster, but there is no guarantee that the post will continue to spread after it reaches its peak. If a post has more friends initially and fewer shares, the post will spread more slowly, but it is more likely to continue to spread for a longer period of time.

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In response to the question, we may say that  Therefore, the inverse of the given matrix is: [ 0.0 1.5 -1.0 ], [ 1.0 -2.0 2.0 ] and [-1.0 1.5 -2.0 ]

An arrangement of integers in rows and columns is known as a matrix. learns about matrices' components and dimensions and first presents them. A matrix is a rectangular grid with integers arranged in rows and columns. For example, Matrix A contains two rows and three columns. The origin of its moniker is due to its single row and 1 n row matrix arrangement. For example, a row matrix of order 1 by 4 is A = [1 2 4 5]. The row matrix P = [-4 -21 -17] of order 1-by-cubic is another example.

Let's begin by calculating the determinant of the matrix:

[tex]det(A) = 1(41 - 30) - 2(40 - 20) + 3(10 - 21)\\det(A) = 4 - 0 - 6\\det(A) = -2\\[/tex]

Since the determinant is not zero, an inverse exists. Let's find the adjugate matrix:

[tex]adj(A) = [dof(A)]^T\\[/tex]

where dof(A) is the matrix of cofactors of A.

[tex]dof(A) = [(-1)^(1+1)0, (-1)^(1+2)2, (-1)^(1+3)(-2),\\(-1)^(2+1)(-3), (-1)^(2+2)4, (-1)^(2+3)(-3),\\(-1)^(3+1)2, (-1)^(3+2)(-4), (-1)^(3+3)*4]\\dof(A) = [0, -2, 2, 3, 4, -3, 2, -4, 4]\\[/tex]

Therefore,

[tex]adj(A) = [0 -3 2; -2 4 -4; 2 -3 4]\\[/tex]

Finally, we can find the inverse matrix by multiplying the adjugate matrix by the reciprocal of the determinant:

[tex]A^(-1) = (1/det(A)) * adj(A)\\A^(-1) = (-1/2) * [0 -3 2; -2 4 -4; 2 -3 4]\\A^(-1) = [0.0 1.5 -1.0; 1.0 -2.0 2.0; -1.0 1.5 -2.0]\\[/tex]

Therefore, the inverse of the given matrix is:

[ 0.0 1.5 -1.0 ]

[ 1.0 -2.0 2.0 ]

[-1.0 1.5 -2.0 ]

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Answer

[ 1.25   -0.75

-0.5      0.5 ]

Step-by-step explanation:

By answering the above question, we may state that As a result, there equation were 108 scouts who attended the jamboree.

A math equation is a mechanism for connecting two claims by using the equals sign (=) to indicate equivalence. A mathematical statement that establishes the equivalence of two mathematical expressions is known as an equation in algebra. The equal symbol, for example, splits the numbers 3x + 5 and 14. A mathematical formula can be used to describe the relationship between two phrases written on opposite sides of a letter. The logo and programme are frequently the same. 2x - 4 Equals 2, for example.  

Because each tin is divided by four scouts, the number of fruit tins necessary will be one-fourth of the number of scouts. As a result, the quantity of tins of fruit required is x/4.

The total number of tins opened, according to the query, is 117. Based on the preceding data, we can construct the following equation:

x/2 + x/3 + x/4 = 117

To remove the denominators, multiply both sides by 12 (the LCM of 2, 3, and 4).

6x + 4x + 3x = 1404

When we simplify, we get:

13x = 1404

By multiplying both sides by 13, we get:

x = 108

As a result, there were 108 scouts who attended the jamboree.

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Answer:

  (i)  25.83 m

  (ii)  178.9 m

  (iii) 4.592 s

Step-by-step explanation:

You want the greatest height, the range, and the time of flight of a particle launched at 45 m/s at an angle of arcsin(0.5) from the horizontal.

The vertical speed is the speed in the direction of launch multiplied by the sine of the angle from the horizontal. Here, it is ...

  (45 m/s)(sin(arcsin(0.5))) = 22.5 m/s

The maximum height is the height at which all of the initial vertical kinetic energy is converted to potential energy:

  h = v²/(2g)

  h = (22.5 m/s)²/(2·9.8 m/s²) ≈ 25.83

The maximum height is about 25.83 meters.

The time of flight is twice the time it takes to reach the maximum height. That time is the height divided by the average vertical speed, which average is half the initial vertical speed.

  T = 2 × (25.83 m)/(11.25 m/s) ≈ 4.592 s

The time of flight is about 4.592 seconds.

The range is the product of the time of flight and the horizontal speed. The horizontal speed is the speed in the launch direction multiplied by the cosine of the angle from the horizontal.

  cos(α) = √(1 -sin(α)²) = √(1 -(1/2)²) = √(3/4)

  R = (45 m/s)√(3/4)·4.592 s = 178.949 m

The horizontal range of the particle is about 178.9 meters.

Therefore , the solution of the given problem of angle comes out to be the particle's horizontal range is approximately 178.9 metres.

A tilt is a form in Euclidean geometry made up of two sides, but rather cylindrical faces, that divide at the barrier's centre and apex. At their junctions, two rays may combine to form an angle. Angle is another outcome of two entities interacting. Dihedral shapes best describe them. A two-dimensional curve can be created by arranging two line beams in various configurations at their ends.

Here,

Lateral momentum

The launch speed is compounded by the sine of the angle from the horizontal to obtain the vertical speed. This is it...

=> 22.5 m/s = (45 m/s)(sin(arcsin(0.5)))

I The tallest person

The height at which all of the original vertical kinetic energy is transformed into potential energy is known as the maximum height.

=> h = v²/(2g) (2g)

=> h = (22.5 m/s)²/(2·9.8 m/s²) ≈ 25.83

The highest point is roughly 25.83 metres high.

(ii) Flight time

The duration of flying is twice as long as the time required to ascend to the highest point. The height divided by the average vertical speed, which is half of the original vertical speed, yields that time.

=> T = 2 × (25.83 m)/(11.25 m/s) ≈ 4.592 s

The journey lasts for approximately 4.592 seconds.

=> (iii) Scope

The range is calculated by multiplying the flight duration by the horizontal speed. The cosine of the angle from the horizontal compounded by the speed in the direction of launch gives the horizontal speed.

=>Cos(x) = (1 - sin(x)2) = (1 -(1/2)2) = Cos(x) = (3/4)

=> R = (45 m/s)√(3/4)·4.592 s = 178.949 m

The particle's horizontal range is approximately 178.9 metres.

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Answer:

33

Step-by-step explanation:

First, you multiply 3 into the other equation (remember, anytime there is a parentheses by each like there is shown, it will be multiplication.) So -3 * -7 is 21 because a negative and a negative is a positive. Then -3 * -4 which is 12. Add 21 and 12 together and you get 33!

I hoped this helped <3

Answer:

Given:

Initial velocity, u = 0 (since the car is stopped)

Final velocity, v = 80 m/s

Time taken, t = 5 s

Required:

Acceleration, a

Equation:

We can use the following equation to calculate the acceleration:

a = (v - u) / t

where u is the initial velocity, v is the final velocity, and t is the time taken.

Solution:

Substituting the given values, we get:

a = (80 - 0) / 5

a = 16 m/s²

Answer:

The acceleration of the car is 16 m/s².

(^2 is just another version of ²)

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a. If Kayla runs on the inside she would run for about 400.96 meters

b. If Emi runs on the outer edge, in one lap, she would run for about 463.76 meters

c. The difference = 62.80 m

This is = 200 + (2π * 42 / 2) + (2π * 42 / 2)

= 200 + 84 π

= 200 + 84 * 3.14

= 463.76 meters

Next we have to solve for the perimeter of the inside edge

200 + (2 π * 32 / 2) +  (2 π * 32 / 2)

= 200 + 64π

= 400.96

The difference is given as 463.76 - 400.96

= 62.80 m

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The identity of cos (2π-x) = -cosx

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

for example;

cos 2 θ + sin 2 θ = 1

1 + cot 2 θ = csc 2 θ

1 + tan 2 θ = sec 2 θ

For cos(2π -x)

π = 180°

2π = 2× 180

= 360°

Therefore cos ( 360-x) will have the identity -cos x.

This is because 360° has an equivalent as 0 in trigonometry.

Therefore cos( 0-x)

= -cosx

therefore the identity of cos (2π-x) is -cos(x).

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The distance on the map where the scale is 1 inch to 40 miles is 1.75 inches.

First we know that the cities are at 2.8 inches apart on a map whose scale is 1 inch to 25 miles, then the real distance between the two cities is:

[tex]D = (2.8)\times25\text{mi}= 70 \ \text{mi}[/tex]

Now we move to a map where each inch is 40 miles, to find the distance in inches, we can take the quotient between the real distance and the scale, it gives:

[tex]D = (\dfrac{70\text{mi}}{40\text{mi}} ) \ \text{inches}[/tex]

[tex]D = 1.75 \ \text{inches}[/tex]

The distance on the other map is 1.75 inches.

The inequality 5.5+37.93d ≤ 300  should be used to calculate the number of days.

The relationship between two non-equal expressions, denoted by a symbol such as 'not equal to, 'greater than' or 'less than' is known as inequality.

She drives 50 miles and per mile, it costs $0.11.

So, for 50 miles, cost = 50 x $0.11 = $5.5

and per day cost = $37.93

Thus, the inequality that must be used to calculate the number of days is

5.5+37.93d ≤ 300 .

the complete question is-

A rental car company charges $37.93 per day to rent a car and $0.11 for every mile driven. Taylor wants to rent a car, knowing that:

She plans to drive 50 miles.

She has at most $300 to spend.

Which inequality can be used to determine dd, the maximum number of days Taylor can afford to rent for while staying within her budget?

5.5d + 37.93 ≥ 300

5.5 + 37.93d ≤300

5.5 + 37.93d ≥ 300

5.5d + 37.93 ≤ 300

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Step-by-step explanation:

Every-other-month for two years ( 24 months)  is  12 deductions

12 * $ 160 = $ 1920

Answer:

range = 60

Step-by-step explanation:

the range of the data set is the difference between the largest and smallest values in the set.

the largest value = 140 ( denoted by the line on furthest right )

the smallest value = 80 ( denoted by line on left )

range = 140 - 80 = 60

The equation that represents the situation is y = x + 6.

An equation is a mathematical statement that shows the equality between two expressions. It consists of two sides separated by an equal sign (=) and can involve numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

The equation that represents the situation is B. y = x + 6.

The problem states that the total cost, y, is the book price, x, plus a $6 shipping fee. We can express this as:

y = x + 6

Option A, y = 6x, suggests that the total cost is only determined by the book price multiplied by 6, which is not what the problem states.

Option C, y = , is not a valid equation as it does not include any terms to determine the value of y.

Option D, y = x - 6, suggests that the total cost is the book price minus $6, which is not what the problem states either.

Therefore, the equation that represents the situation is y = x + 6.

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The initial number of cups in Box B at first is given as follows:

810 cups.

The initial number of cups in box B is obtained solving a system of equations, for which the variables A, B and C represents the amounts of each box.

A shop owner had three boxes, containing a total of 1500 plastic cups. The number of plastic cups in Box A to the total number of plastic cups was 3:10, hence the initial amount of Box A is given as follows:

0.3 x 1500 = 450 cups.

Then:

B + C = 1500 - 450

B + C = 1050.

He sold 330 plastic cups from Box B and sold 1/3 of the plastic cups in  = Box C. The number of plastic cups left in Box B to the number of plastic cups left in Box C was 3:1, hence:

(B - 330)/(2C/3) = 3

B - 330 = 2C

C = 0.5B - 165

Replacing into B + C = 1050, the value of B is given as follows:

B + 0.5B - 165 = 1050

B = 1215/1.5

B = 810.

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To determine which company would give Amy a stable long-term investment, we need to compare the standard deviations of the stock prices for both companies. A lower standard deviation indicates that the stock price is more stable and has less variability.

From the given information, we can see that the standard deviation of stock prices for Garden Statues Express is $1.05, while the standard deviation of stock prices for Masterful Pocketwatches is $9.83. This means that Garden Statues Express has a much lower standard deviation, and thus a more stable stock price, compared to Masterful Pocketwatches.

Therefore, if Amy is looking for a stable long-term investment, she should consider investing in Garden Statues Express.

The expressions for the dimensions of the rectangular composite figure, indicates;

a) The simplified expression for the shaded region is; (3·x + 36) cm²

b) x = 17

c) Please find attached the drawing of the figure, showing the dimensions, created with MS Word

d) The area of the shaded region cannot be 39 cm², because that will make the length of the small rectangle -1 cm.

A composite figure is a figure that comprises of two or more simpler figures.  

The figure can be considered as a composite figure, comprising of a small rectangular unshaded region within a shaded larger rectangle.

The area of the shaded region is the difference between the area of the large rectangle and the area of the unshaded rectangle, which can be found as follows;

Area of a rectangle = Length × Breadth

Length of the large rectangle = (x + 5) cm

Breadth of the large rectangle = 6 cm

Area of large rectangle = 6 cm × (x + 5) cm = (6·x + 30) cm²

Area of the unshaded area = 3 cm × (x - 2) cm = (3·x - 6) cm²

Area of the shaded region, A, is therefore;

A = (6·x + 30) cm² - (3·x - 6) cm² = (3·x + 36) cm²

b) The shaded area = 87 cm², therefore;

(3·x + 36) = 87

3·x = 87 - 36 = 51

x = 51/3 = 17

c) The dimensions of the figures are therefore;

x = 17

Please find attached the drawing of the shape created with MS Word

d) The area of the shaded region is; (3·x + 36)

Therefore, if the area of the shaded region is 39, we get;

(3·x + 36) = 39

x = (39 - 36)/3 = 1

x = 1

The length of the unshaded region is; (x - 2)

The length of the unshaded region becomes; (1 - 2) = -1, which is not a natural number and therefore, not possible

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Answer:

Starting with the expression n² - 2 - (n - 2)², we can simplify it by expanding the second term using the formula for the square of a binomial:

n² - 2 - (n - 2)² = n² - 2 - (n² - 4n + 4)

Next, we can combine like terms:

n² - 2 - (n² - 4n + 4) = n² - n² + 4n - 6

Simplifying further, we get:

n² - 2 - (n - 2)² = 4n - 6

To prove that this expression is always even, we can show that it can be written in the form 2k, where k is an integer.

So, let's rewrite the expression as:

4n - 6 = 2(2n - 3)

Since 2n - 3 is an integer (because n is an integer greater than 1), we have shown that 4n - 6 can be written in the form 2k, where k is an integer.

Therefore, we have proved algebraically that n² - 2 - (n - 2)² is always an even number.