How much will the monthly payment be for a new car priced at $29,950 if the current finance rate is 36 months at 3. 16%? Include financing the 8% TT&L and make a 25% down payment.



I need the answer fast!!

Answer:

y = |x - 1|

Step-by-step explanation:

This is an absolute value function in the form g(x) = |x - h| + k, where

(h, k) is the vertex, (1, 0).  Substitute these values into the function to get the equation of the graph:

y = |x - 1| + 0 = |x - 1|

(a) y(x) = c1 e^(-4x/3) cos(2x) + c2 e^(-4x/3) sin(2x), stable node at the origin;

(b) y(x) = c1 e^(2x) + c2 e^(-4x), unstable node at the origin;

(c) y(x) = c1 e^(-x/22) cos(sqrt(119)x/22) + c2 e^(-x/22) sin(sqrt(119)x/22), stable node at the origin;

(d) y(x) = c1 e^(sqrt(3)x/2) + c2 e^(-sqrt(3)x/2), unstable saddle at the origin;

(e) y(x) = c1 e^x cos(2x) + c2 e^x sin(2x), stable spiral at the origin;

(f) y(x) = c1 cos(3x/2) + c2 sin(3x/2), stable limit cycle around the origin;

(g) y(x) = c1 e^(-x/3) + c2 e^(-x), stable node at the origin.

(a) The characteristic equation is 2r^2 + 9r + 4 = 0, with roots r1 = -4/3 and r2 = -1/2. The general solution is y(x) = c1 e^(-4x/3) cos(2x) + c2 e^(-4x/3) sin(2x). The equilibrium at the origin is a stable node since both eigenvalues have negative real parts.

(b) The characteristic equation is r^2 + 2r - 8 = 0, with roots r1 = 2 and r2 = -4. The general solution is y(x) = c1 e^(2x) + c2 e^(-4x). The equilibrium at the origin is an unstable node since both eigenvalues have positive real parts.

(c) The characteristic equation is 44r^2 - 12r + 5 = 0, with roots r1 = (3 + sqrt(119))/22 and r2 = (3 - sqrt(119))/22. The general solution is y(x) = c1 e^(-x/22) cos(sqrt(119)x/22) + c2 e^(-x/22) sin(sqrt(119)x/22). The equilibrium at the origin is a stable node since both eigenvalues have negative real parts.

(d) The characteristic equation is 2r^2 - 3 = 0, with roots r1 = sqrt(3)/2 and r2 = -sqrt(3)/2. The general solution is y(x) = c1 e^(sqrt(3)x/2) + c2 e^(-sqrt(3)x/2). The equilibrium at the origin is an unstable saddle since the eigenvalues have opposite signs.

(e) The characteristic equation is r^2 - 2r + 5 = 0, with roots r1 = 1 + 2i and r2 = 1 - 2i. The general solution is y(x) = c1 e^x cos(2x) + c2 e^x sin(2x). The equilibrium at the origin is a stable spiral since both eigenvalues have negative real parts and non-zero imaginary parts.

(f) The characteristic equation is 4r^2 + 9 = 0, with roots r1 = 3i/2 and r2 = -3i/2. The general solution y(x) = c1 cos(3x/2) + c2 sin(3x/2), stable limit cycle around the origin.

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The surface area of revolution of C about the x-axis is π/27 (81^(3/2) - 1) or approximately 478.48 units².

To find the surface area of revolution of C about the x-axis, we can use the formula:

Surface area = ∫2πy ds

where y is the function that defines the curve C, and ds is an element of arc length along the curve.

We can express ds in terms of dx as follows:

ds = √(1 + (dy/dx)²) dx

where dy/dx is the derivative of y with respect to x.

For the curve C, we have:

y = 3x³

dy/dx = 9x²

Substituting these into the expression for ds, we get:

ds = √(1 + (9x²)²) dx

= √(1 + 81x⁴) dx

Substituting y and ds into the formula for surface area, we get:

Surface area = ∫₂πy √(1 + (dy/dx)²) dx

= ∫₀³ 2π(3x³) √(1 + 81x⁴) dx

This integral can be evaluated using substitution:

Let u = 1 + 81x⁴

Then du/dx = 324x³

And dx = du/324x³

Substituting these into the integral, we get:

Surface area = ∫₁₀³ 2π(3x³) √(1 + 81x⁴) dx

= 2π/108 ∫₁₀³ (3x³) √u du

= π/54 ∫₁₀³ u^(1/2) du

= π/54 (2/3) u^(3/2) | from 1 to 81

= π/81 (2/3)(81^(3/2) - 1)

= π/27 (81^(3/2) - 1)

Therefore, To find the surface area of revolution of C about the x-axis, we can use the formula:

Surface area = ∫2πy ds

where y is the function that defines the curve C, and ds is an element of arc length along the curve.

We can express ds in terms of dx as follows:

ds = √(1 + (dy/dx)²) dx

where dy/dx is the derivative of y with respect to x.

For the curve C, we have:

y = 3x³

dy/dx = 9x²

Substituting these into the expression for ds, we get:

ds = √(1 + (9x²)²) dx

= √(1 + 81x⁴) dx

Substituting y and ds into the formula for surface area, we get:

Surface area = ∫₂πy √(1 + (dy/dx)²) dx

= ∫₀³ 2π(3x³) √(1 + 81x⁴) dx

This integral can be evaluated using substitution:

Let u = 1 + 81x⁴

Then du/dx = 324x³

And dx = du/324x³

Substituting these into the integral, we get:

Surface area = ∫₁₀³ 2π(3x³) √(1 + 81x⁴) dx

= 2π/108 ∫₁₀³ (3x³) √u du

= π/54 ∫₁₀³ u^(1/2) du

= π/54 (2/3) u^(3/2) | from 1 to 81

= π/81 (2/3)(81^(3/2) - 1)

= π/27 (81^(3/2) - 1)

Therefore, the surface area of revolution of C about the x-axis is π/27 (81^(3/2) - 1) or approximately 478.48 units².

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The total budget of the prom, using the percentage concept, is given by = $ 4000.

Hence the correct option is (D).

Let the total budget be $ 100x.

Spending on Entertainment is 25% of the Budget.

So the spending on Entertainment = 100x*(25/100) = $ 25x.

Spending on Refreshments is 30% of the Budget.

So the spending on Refreshments = 100x*(30/100) = $ 30x.

Spending on Decorations is 45% of the Budget.

So the spending on Decorations = 100x*(45/100) = $ 45x.

It is also given that the spending on refreshments in dollar is $ 1200.

According to information,

30x = 1200

x = 1200/30

x = 40

Hence the total budget is = $100x = $ 100*40 = $ 4000.

So the correct option is (D).

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The question is incomplete. The complete question will be -

"The student council is planning for the prom. They have broken down their expenses in the graph below. If they decide they will spend $1,200 on refreshments, then what is their total budget?

Prom Budget:

Entertainment: 25%

Refreshments: 30%

Decorations: 45%

Answer choices:

A. $3,600

B. $6,000

C. $7,200

D. $4,000"

A comparison of the cost of each for 100 g indicates that the standard which costs about $1.13 per 100 grams is more expensive than the supersize which costs about $0.69 per 100 g and the supersize is more expensive than the economy which costs $0.575 per 100 g.

The order of the cost of the brands from cheapest to most expensive can be presented as follows;

The cost per unit is the cost of a number of items, divided by the number of units of the item.

The unit cost of the brands of cereal sold at the supermarket are therefore;

Unit cost per 100 gram for Standard = $2.49/220 g × 100 ≈ $1.13/g

Unit cost per gram for Economy = $5.00/725 g × 100 = $100/145/ g ≈ $0.69/g

Unit cost per gram for Supersize = $11.50/2kg × 100 = $1150/2,000 g ≈ $0.575/g

Based on the above unit cost, we get;

1.13 > 0.69 > 0.575

Therefore, the cheapest brand is the supersize with a cost per 100 gram of $0.575/g, followed by the economy, with a cost per 100 grams which is about $0.69/g then the most costly is the standard, with a cost per 100 gram of $1.13/g

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Answer: 4.7KL

Step-by-step explanation:

KL=DAL/100

KL=470/100

KL=4.7

There are several steps a consumer market can take to become more knowledgeable like research and asking questions.

Research The first step is to  probe the product or service that you're interested in. This involves looking at reviews, product descriptions, and comparing prices. You can also look for information from  dependable sources  similar as consumer reports or government websites.   Ask questions If you have any  dubieties or  enterprises, don't  vacillate to ask the  dealer or service provider.

Ask them about their experience and qualifications, and make sure to clarify any terms or conditions that are unclear.   Get a alternate opinion If you're  doubtful about a product or service, seek the advice of someone you trust or who has  moxie in that area. They can help you make an informed decision grounded on their knowledge and experience.

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Therefore, the expression 250 * [tex]2^h[/tex] gives the number of bacteria in the colony after h hours, assuming that the initial number of bacteria is 250.

In mathematics, an expression is a combination of numbers, variables, and mathematical operations, such as addition, subtraction, multiplication, and division. Expressions can also include parentheses, brackets, and other symbols to clarify the order of operations. Expressions can be evaluated to obtain a numerical result or can be simplified using mathematical rules and properties to make them easier to work with. In algebra, expressions are often used to represent relationships between variables and to solve equations and inequalities.

Here,

In the given expression 250*[tex]2^h[/tex], the constant 250 represents the initial number of bacteria in the colony.

Since the number of bacteria doubles every hour, if we start with 250 bacteria at the beginning, after one hour we will have 250 * 2 = 500 bacteria. After two hours, we will have 500 * 2 = 1000 bacteria, and so on.

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Raymond is 1984 years old and Alvin is 16 years old.

Let's represent Raymond's age with x and Alvin's age with y.

According to the problem, we have the following two equations:

x + y^2 = 2240 (equation 1)

y + x^2 = 1008 (equation 2)

We can solve this system of equations by substituting one equation into the other to eliminate one of the variables. Let's solve equation 1 for x:

x = 2240 - y^2

Now we substitute this expression for x into equation 2:

y + (2240 - y^2)^2 = 1008

Simplifying and solving for y:

y + 5017600 - 4480y^2 + y^4 = 1008

y^4 - 4480y^2 + y + 5016592 = 0

We can use a numerical solver or factorization to find the solutions. By inspection, we can see that y = 16 is a solution (16 + 1008 = 1024, which is a perfect square).

Now we can use synthetic division to factor out (y - 16) from the polynomial:

16 | 1 0 -4480 1 5016592

16 2560 -35760 -358592

1 16 -1920 -35759 4658000

So we have:

(y - 16)(y^3 + 16y^2 - 1920y - 35759) = 0

We can use a numerical solver or synthetic division again to find the other solutions, but by inspection we can see that the cubic factor has only one real root, which is approximately -19.103. Therefore, we have:

y = 16, x = 2240 - y^2 = 2240 - 256 = 1984

So Raymond is 1984 years old and Alvin is 16 years old.

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Answer:

g(x) = -2f(x)

Step-by-step explanation:

From the graph, we can see that g(x) is a reflection of f(x) about the x-axis, followed by a vertical stretch by a factor of 2. This is equivalent to multiplying f(x) by -2, which gives us the transformation:

g(x) = -2f(x)

To solve this problem, we can use something called the law of sines. This is a proportional relationship in which the sine of one angle over the opposite side is equal to the sine of another angle over its opposite side.

sin(a) / a = sin(b) / b = sin(c) / c

To use the law of sines, we will need to figure out the measure of angle C, however.

27 + 132 + C = 180

159 + C = 180

C = 21

Now that we have sides and their opposite angles, we can apply the law of sines.

sin(27) / AC = sin(21) / 26

AC x sin(21) = sin(27) x 26

AC = [ sin(27) x 26 ] / sin(21)

AC = 32.9375

AC (rounded) = 32.9

Answer: AC = 32.9 m

Hope this helps!

The transformation appears to be a rigid motion because A. Yes, because the angle measures and the distances between the vertices are the same as the corresponding angle measures and distances in the preimage.

A rigid motion transformation, colloquially referred to as an isometry, preserves the conformation and magnitude of a geometric construct. This change consists of translations, rotations, and reflections.

For this particular example, the preimage happens to be a right triangle facing leftward, whilst the image is an inverted right triangle facing eastward. This transmutation can be realized through a conjunction of reflection and rotation while maintaining similar angle measurements and distances between vertices. As a result, it is evidently a rigid motion.

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The length of the missing segment is given as follows:

? = 4.4.

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The theorem is expressed as follows:

c² = a² + b².

In which:

The hypotenuse length for the right triangle is given as follows:

h² = 6.6² + 8.8²

[tex]h = \sqrt{6.6^2 + 8.8^2}[/tex]

h = 11.

The hypotenuse segment is divided into a radius of 6.6 plus the missing segment of ?, thus:

6.6 + ? = 11

? = 11 - 6.6

? = 4.4.

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The radius of the circle is 5.

To complete the proof, we need to find the radius of the circle centered at the origin and containing the point (5, 0). We can use the distance formula to find the distance between the origin (0, 0) and the point (5, 0):

distance = √((5 - 0)^2 + (0 - 0)^2) = √25 = 5

Therefore, the radius of the circle is 5.

Now, to determine whether the point (, −3) lies on the circle, we need to find the distance between the origin and the point (, −3):

distance = √((-3 - 0)^2 + (0 - 0)^2) = √9 = 3

Since the distance between the origin and the point (, −3) is not equal to the radius of the circle, which is 5, we can conclude that the point (, −3) does not lie on the circle centered at the origin and containing the point (5, 0).

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The volume of the given cylinder is 84.78 cubic yards.

The first step to finding the volume of a cylinder is to use the formula V = πr^2h, where r is the radius of the cylinder (which is half of the diameter). So, to find the radius, we divide the diameter (which is 3 yards) by 2, giving us a radius of 1.5 yards. Then, we can plug in the values for radius (1.5), height (12), and π (3.14) into the formula to find the volume:

V = πr^2h
V = 3.14 x 1.5^2 x 12
V = 84.78 cubic yards

Therefore, the volume of the cylinder is 84.78 cubic yards.
Hi! To find the volume of a cylinder with a diameter of 3 yards and a height of 12 yards, the first step is to find the radius.

Step 1: Since the diameter is 3 yards, you can find the radius by dividing the diameter by 2. Radius = Diameter / 2. So, the radius is 1.5 yards.

Step 2: Now, you can find the volume of the cylinder using the formula: Volume = π × (radius^2) × height. In this case, Volume = π × (1.5^2) × 12.

Step 3: Calculate the volume: Volume = π × 2.25 × 12. Using the value of π as approximately 3.14, the volume becomes 3.14 × 2.25 × 12.

Step 4: Multiply the numbers: Volume ≈ 84.78 cubic yards.

So, the volume of the cylinder is approximately 84.78 cubic yards.

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To maximize Q = xy with the constraint x + y = 332, and given x = 4, we need to express the objective function in terms of y.

Since x = 4, we can rewrite the constraint as: 4 + y = 332

Now, solve for y:
y = 332 - 4
y = 328

Now, substitute the value of x into the objective function:
Q = (4)(y)

So, the objective function in terms of y is:
Q = 4y

To write the objective function in terms of y, we can solve for x in the constraint equation:

x + 332 = 4

x = 4 - 332

x = -328

Now we can substitute this value of x into the equation for Q:

Q = xy

Q = (-328)y

Q = -328y

Therefore, the objective function in terms of y is Q = -328y.

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To find the area enclosed by the figure, we first need to find the area of  the square and the  area of each semicircle.

The area of the square is simply the length of one of its sides squared, which is:

54 m x 54 m = 2,916 m²

The area of each semicircle is half the area of a full circle with the same radius as the side of the square. The radius of each semicircle is 54 m/2 = 27 m.

The area of each semicircle is:

1/2 x π x 27 m² = 1/2 x 3.14 x 27 m x 27 m ≈ 1,442.31 m²

Since there are four semicircles, the total area of the semicircles is:

4 x 1,442.31 m² = 5,769.24 m²

Therefore, the total area enclosed by the figure is:

2,916 m² + 5,769.24 m² ≈ 8,685.24 m²

Frank's answer of 1,662.12 m² is significantly less than the actual area. He may have made the mistake of only calculating the area of one of the semicircles instead of all four, or he may have forgotten to include the area of the square.

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If the mold is not a rectangular prism, then the formula for its volume will be different, depending on its shape.

To calculate the volume of the wooden mold, we need to know its dimensions. Let's assume that the mold is in the shape of a rectangular prism, with length (L), width (W), and height (H).

Then, the volume (V) of the wooden mold is given by the formula:

V = L x W x H

To find the values of L, W, and H, the construction company needs to measure the dimensions of the mold using a measuring tape or ruler. Once they have the measurements, they can plug them into the formula above to find the volume of the mold.

It's important to note that the units of measurement used for the dimensions of the mold should be consistent. For example, if the length and width are measured in feet, then the height should also be measured in feet, and the resulting volume will be in cubic feet.

If the mold is not a rectangular prism, then the formula for its volume will be different, depending on its shape.

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The length of the rectangle is 8 feet given it is 7 feet wide and has an area of 56 square feet.

A rectangle is a geometric figure with four straight sides and four right angles, where the opposite sides are parallel and equal in length. In this case, we are given that the rectangle has a width of 7 feet and an area of 56 square feet. The area of a rectangle is calculated by multiplying its length (L) and width (W), expressed as A = L × W.

We are provided with the width, W = 7 feet, and the area, A = 56 square feet. To find the length of the rectangle, we can rearrange the area formula:

L = A ÷ W

Substituting the given values, we have:

L = 56 ÷ 7

L = 8 feet

Hence, the length of the rectangle is 8 feet. To summarize, a rectangle with a width of 7 feet and an area of 56 square feet has a length of 8 feet. The dimensions of the rectangle are 7 feet by 8 feet.

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The probability of someone getting the mac and cheese or fries, with a burger and a chocolate cupcake is 1/6.

To determine the probability of someone getting the mac and cheese or fries, with a burger and a chocolate cupcake, we need to look at the possible combinations and find the ones that meet these criteria.

There are 3 side options, 2 sandwich options, and 2 dessert options, making a total of 3 x 2 x 2 = 12 possible combinations.

Now let's find the combinations that fit the desired meal:
1. Mac and cheese, burger, chocolate cupcake
2. Fries, burger, chocolate cupcake

There are 2 favorable combinations. Therefore, the probability is:

2 (favorable combinations) / 12 (total combinations) = 1/6

So, the probability of someone getting the mac and cheese or fries, with a burger and a chocolate cupcake is 1/6.

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The probability of getting a survey of size n = 115, where 56 or fewer subjects say they feel financially comfortable is 0.0003. Therefore, the correct option is B.

To find the probability of getting a survey of size n = 115, where 56 or fewer subjects say they feel financially comfortable if the residents' economic opinions have not changed from last year, we can use the normal distribution as an approximation to the binomial distribution.

1. Calculate the mean (μ) and standard deviation (σ) of the binomial distribution using the proportion from the census (64%) and the sample size (n = 115).

Mean (μ) = n * p = 115 * 0.64 = 73.6

Standard deviation (σ) = √(n * p * (1-p)) = √(115 * 0.64 * 0.36) ≈ 6.45

2. Convert the observed value (56) to a z-score:

z = (x - μ) / σ = (56 - 73.6) / 6.45 ≈ -2.73

3. Find the probability of getting a z-score less than or equal to -2.73 using a z-table or an online calculator.

P(Z ≤ -2.73) ≈ 0.0032

The closest answer choice to this probability is 0.0003, so the correct answer is (b) 0.0003.

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iStar Financial (STAR) has a lower PE ratio than Bank Rate (RATE), which suggests that it may be financially stronger

To determine which company is financially stronger using their PE ratios, we need to calculate the PE ratio for each company. PE ratio, or price-to-earnings ratio, is a financial metric used to measure the valuation of a company's stock. It is calculated by dividing the market price per share by the earnings per share.

For Bank Rate (RATE), the PE ratio can be calculated as:

PE ratio = market price per share / earnings per share

PE ratio = $13.95 / $2.71

PE ratio = 5.14

For iStar Financial (STAR), the PE ratio can be calculated as:

PE ratio = market price per share / earnings per share

PE ratio = $12.18 / $3.62

PE ratio = 3.37

A lower PE ratio indicates that a company's stock is relatively undervalued compared to its earnings. In this case, iStar Financial (STAR) has a lower PE ratio than Bank Rate (RATE), which suggests that it may be financially stronger.

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Direction of maximum rate of increase: (-14, 3, 7/3).

The rate of change in that direction:√(196 + 9 + 49/9).

To determine the direction in which f has a maximum rate of increase from point P(-1, 7, 9):

We need to find the gradient of the function f(x, y, z) = x²y√z.

The gradient is given by the vector of partial derivatives with respect to x, y, and z:

∇f = (df/dx, df/dy, df/dz)

First, find the partial derivatives:

df/dx = 2xy√z
df/dy = x²√z
df/dz = (1/2)x²y*z^(-1/2)

Now, evaluate the gradient at point P(-1, 7, 9):

∇f(P) = (2(-1)(7)√9, (-1)²√9, (1/2)(-1)²(7)*(9^(-1/2)))
∇f(P) = (-14, 3, 7/3)

The direction of maximum rate of increase is given by the gradient at point P, which is (-14, 3, 7/3).

To determine the rate of change in that direction:

The rate of change is given by the magnitude of the gradient vector:

Rate of change = ||∇f(P)|| = √((-14)^2 + (3)^2 + (7/3)^2)
Rate of change = √(196 + 9 + 49/9)

The rate of change is the square root of this value, which is an exact representation of the rate of change in the direction of maximum increase.

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The approximate APR for James' cabinet purchase can be calculated using the formula Approximate APR = (Finance Charge ÷ #Months) (12) ÷ Amount Financed. Plugging in the given values, we get (49 ÷ 18) (12) ÷ 438 = 0.0397 or 3.97%. Rounded to the nearest tenth, the approximate APR is 4%.

APR, or Annual Percentage Rate, is the annual interest rate charged by a lender for borrowing money. It includes not only the interest rate but also any additional fees or charges associated with the loan. The APR helps borrowers compare different loan offers and understand the true cost of borrowing.

It is important to note that the APR is an approximation and may differ from the actual interest rate charged over the life of the loan, especially if the loan has variable rates or fees. When considering a loan, it is important to compare not just the APR but also the terms and conditions of the loan, such as the repayment period, monthly payments, and any penalties for early repayment.

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The polar coordinates (r, θ) for point B with rectangular coordinates (-5, 12) are (13, 112.62°).

The polar coordinates (r, θ) for point B with rectangular coordinates (-5, 12) can be determined as follows.

1. Calculate the radius r:

r = √(x² + y²) = √((-5)² + 12²) = √(25 + 144) = √169 = 13.

2. Calculate the angle θ in radians:

θ = arctan(y/x) = arctan(12/-5) ≈ -1.176 radians.

3. Convert θ from radians to degrees:

θ = (-1.176 * 180) / π ≈ -67.38 degrees.

4. Adjust the angle to the proper quadrant (since point B is in the second quadrant):

θ = 180 - 67.38 = 112.62 degrees.

So, the polar coordinates (r, θ) for point B are (13, 112.62°).

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A collection of information, such as numbers, measurements, or observations, is commonly referred to as data.

Data can be collected from a wide variety of sources, such as surveys, experiments, or observations, and can be analyzed to reveal patterns, trends, and relationships.

Data can be represented in many different ways, including tables, graphs, charts, or statistical summaries, and can be used to make informed decisions in a wide range of fields, including business, science, healthcare, and social sciences.

However, it is important to ensure that the data is accurate, reliable, and unbiased, and that appropriate statistical methods are used to analyze and interpret it.

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Full Question: What is a collection of information such as number measurement or observation called?

We can use the empirical rule to estimate the percentage of workdays it takes Susan between 38 and 46 minutes to drive to work. According to the empirical rule, for a normal distribution:

- About 68% of the data falls within one standard deviation of the mean

- About 95% of the data falls within two standard deviations of the mean

- About 99.7% of the data falls within three standard deviations of the mean

Since the mean is 42 minutes and the standard deviation is 4 minutes, one standard deviation below the mean is 38 minutes, and one standard deviation above the mean is 46 minutes. So, about 68% of the time it takes Susan between 38 and 46 minutes to drive to work.

Therefore, the answer is (B) 68%.

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The value of the record high temperature is 114°F

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

This means that

High - Low = 109

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

High - 5 = 109

Add 5 to both sides

High = 114

Hence, the record high temperature is 114°F

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Let's assume that the length of the rectangle is x and the width is y.

The perimeter of the rectangle is given by:

P = 2x + 2y

Substituting P = 62, we get:

62 = 2x + 2y

Simplifying the above equation, we get:

x + y = 31

The area of the rectangle is given by:

A = xy

To find the area, we need to solve for one of the variables, either x or y. We can use the equation x + y = 31 to solve for y in terms of x:

y = 31 - x

Substituting this expression for y in the equation for the area, we get:

A = x(31 - x)

Expanding the above expression, we get:

A = 31x - x^2

To find the maximum area, we need to find the value of x that maximizes A. We can do this by completing the square:

A = - (x^2 - 31x)

A = - [(x - 15.5)^2 - 240.25]

A = 240.25 - (x - 15.5)^2

The maximum area occurs when x = 15.5, so the dimensions of the rectangle are:

x = 15.5 mm

y = 31 - x = 15.5 mm

The area of the rectangle is:

A = xy = 15.5 mm × 15.5 mm = 240.25 mm^2

Therefore, the area of the rectangle is 240.25 square millimeters.

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