12. A normal distribution has a mean of 34 and a standard deviation of 7. Find the range of
values that represent the middle 95% of the data.

F. 27 G. 20 X 48
H. 13 J. 6

The differential equation that describes the situation is: dp/dt = 41.43 * p * (1 - p/555).

The logistic differential equation is a commonly used model for population growth or decay, taking into account the carrying capacity of the environment. It is given by:

dp/dt = r * p * (1 - p/K)

where p is the population, t is time, r is the growth rate, and K is the carrying capacity.

In this case, the maximum capacity of the island is 555 bears, so we have K = 555. At 6 AM, the number of bears on the island is 165 and is increasing at a rate of 29 bears per day, so we have:

p(0) = 165 and dp/dt(0) = 29

To write the differential equation that describes this situation, we can use the initial conditions and the logistic model:

dp/dt = r * p * (1 - p/555)

Substituting the initial conditions, we get:

29 = r * 165 * (1 - 165/555)

Simplifying this expression, we get:

29 = r * 0.7

r = 41.43

Therefore, the differential equation that describes the situation is:

dp/dt = 41.43 * p * (1 - p/555)

Note that this model assumes that the growth rate of the bear population is proportional to the number of bears present and that the carrying capacity is fixed. Real-life situations may involve more complex models with time-varying carrying capacities or other factors affecting population growth.

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For the Portfolio for Unit 5 Part 1, the Car Wheel Project, you'll want to include several key components.

First, be sure to include a detailed description of the project itself, including any goals or objectives you had in mind when you started. This could include things like improving your engineering or design skills, learning more about the materials used in car wheels, or simply creating a visually impressive final product.

Next, include some documentation of your process as you worked on the project. This might include sketches, diagrams, or photos of different stages of the process. Be sure to highlight any challenges or roadblocks you encountered along the way, and how you overcame them.

Finally, be sure to include a final showcase of your completed car wheel project. This might include photos of the finished product from different angles, a video demonstrating how it works or how it was made, or even a physical prototype that you can bring in to show off.

Overall, the key to a successful portfolio for the Car Wheel Project is to demonstrate your creativity, your technical skills, and your ability to work through challenges and solve problems.

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Unit 5 Portfolio Car Wheel Project Answer the following questions: 1. How do you find the area of a circle?

2. How do you find the circumference of a circle?

3. What information do you need in order to find the area or circumference of a circle

4. To determine the number of full rotations your tires complete before you need to switch thefront and back tires, will you be working with area or circumference?

Answer:

How do you find an area of a crilce?

-Use the formula A=[tex]\pi r^2[/tex]

How do you find the cirumference of a cirlce?

-Use the formula c=2[tex]\pi r[/tex]

What information do you need in order to find the area or circumference of a cirlce?

-I believe it's the diameter.

To determine the number of full rotations your tires complete before you switch the front and back tires will you be working with area or circumference?

-Circumference. Because ifts the distance around something, like a circle, or a wheel.

(My three examples of the circumference of tires)

Bike:

C=2(3.14)r

C=2 (3.14) 8

C=50.24

Car:

C=2(3.14)r

C=2 (3.14) 20

C=125.6

Scooter:

C=2(3.14)r

C=2 (3.14) 3

C=18.24

(My tire rotations how far you'll go in 10,000 miles)

car-

20x3.1416=62.83

10,000/62.83= 159.2

scooter-

3x3.1416=9.4248

10,000/9.4248=1,061

bike-

8x3.1416=25.13

10,000/25.13=397.9

Using one of your vehicles how far can you get in a week?

-We're driving the car with twenty inch wheels and we're going seventy miles an hour. So in a week we would have driven 490 miles.

When will you need to change your tires?

-When we reach 225 miles we'd change the tires. Because these tires are heavier than the average tire they're wear out faster.

I hope this helps!!! Let me know if you need some help with anything else.

To find the other possible vertices of the square, we need to determine the coordinates of the other three vertices. Since we know that the length of each side is 3 units, we can use this information to determine the distance between the given vertex (5, 4) and the other vertices.

First, we can determine the direction of the square by looking at the given vertex and knowing that the sides of a square are equal in length and perpendicular. Since we know that the side length is 3 units, we can move 3 units to the right to find one possible vertex. This gives us the point (8, 4).

Next, we can move 3 units up to find another possible vertex. This gives us the point (5, 7).

Finally, we can move 3 units to the left to find the last possible vertex. This gives us the point (2, 4).

Therefore, the letter that should be circled by all the ordered pairs that could be another vertex is D, which represents the point (2, 4).

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The surface area of the cone is approximately 793.10 cm².

What is cone?

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

What is area of cone?

The surface area of a cone is given by the formula:

Surface Area = πr² + πrℓ

where r is the radius of the circular base and ℓ is the slant height of the cone.

To find the surface area of a cone, we need to find the area of its base and the area of its lateral surface and then add them together.

The formula for the lateral surface area of a cone is given by:

L = πrℓ

where r is the radius of the base and ℓ is the slant height of the cone.

The formula for the area of the base of a cone is given by:

B = πr²

where r is the radius of the base.

Given the slant height ℓ = 19 and the radius r = 9, we can find the lateral surface area of the cone as follows:

L = πrℓ

= π(9)(19)

≈ 538.63 cm²

Next, we can find the area of the base of the cone as follows:

B = πr²

= π(9)²

≈ 254.47 cm²

Therefore, the surface area of the cone is:

A = B + L

≈ 793.10 cm²

So, the surface area of the cone is approximately 793.10 cm².

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The annual interest rate the third party will receive for the investment is approximately 7.7%.

After buying a new car, you sold your old car and took a 180-day note for $4,500 at 7.5% simple interest as payment. The principal plus interest will be due at the end of 180 days.

First, let's calculate the total amount due at the end of the 180 days. To do this, we'll use the formula for simple interest:

Interest = Principal x Rate x Time
Interest = $4,500 x 7.5% x (180/360) [As it's for half a year]
Interest = $4,500 x 0.075 x 0.5
Interest = $168.75

Now, let's add the interest to the principal to find the total amount due:
Total Amount = Principal + Interest
Total Amount = $4,500 + $168.75
Total Amount = $4,668.75

Sixty days later, you sell the note to a third party for $4,550. The third party will receive the remaining interest for 120 days. We need to find the annual interest rate for the third party's investment.

First, let's find the remaining interest:
Remaining Interest = Total Amount - $4,550
Remaining Interest = $4,668.75 - $4,550
Remaining Interest = $118.75

Now, let's find the annual interest rate for the third party's investment using the simple interest formula, but solving for the rate:

Rate = (Remaining Interest) / (Principal x Time)
Rate = $118.75 / ($4,550 x (120/360))
Rate = $118.75 / ($4,550 x 0.3333)

Rate ≈ 0.077 (approximated to 3 decimal places)

So, the annual interest rate the third party will receive for the investment is approximately 7.7%.

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To find the projection of u onto v, we need to use the formula:

projv(u) = (u · v / |v|^2) * v

where u · v is the dot product of u and v, and |v|^2 is the magnitude of v squared.

First, we calculate u · v:

u · v = (-8i - 4j - 2k) · (0i + 2j + 2k) = 0 + (-8*0) + (-4*2) + (-2*2) = -8

Next, we calculate |v|^2:

|v|^2 = (2)^2 + (2)^2 = 8

Now, we can plug these values into the projection formula:

projv(u) = (-8 / 8) * (2j + 2k) = -1j - 1k

So the projection of u onto v is -1j - 1k.

To find the vector component of u orthogonal to v, we can use the formula:

u - projv(u)

We already found projv(u) to be -1j - 1k, so we just need to subtract that from u:

u - projv(u) = (-8i - 4j - 2k) - (-1j - 1k) = -8i - 3j - k

Therefore, the vector component of u orthogonal to v is -8i - 3j - k.
To find the projection of u onto v, we will use the formula:

proj_v(u) = (u•v / ||v||^2) * v

where u = -8i - 4j - 2k, v = 2j + 2k, and • denotes the dot product.

First, calculate the dot product (u•v):

u•v = (-8)(0) + (-4)(2) + (-2)(2) = -8 - 4 = -12

Next, calculate the magnitude squared of v (||v||^2):

||v||^2 = (0^2) + (2^2) + (2^2) = 0 + 4 + 4 = 8

Now, use the formula:

proj_v(u) = (-12 / 8) * v = -1.5 * (2j + 2k) = -3j - 3k

Next, to find the vector component of u orthogonal to v, use the formula:

u_orthogonal = u - proj_v(u)

u_orthogonal = (-8i - 4j - 2k) - (-3j - 3k) = -8i - 1j + 1k

So, the projection of u onto v is -3j - 3k, and the vector component of u orthogonal to v is -8i - 1j + 1k.

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"A recipe for chocolate chip cookies calls for 4/5 cup of sugar per batch. If a baker wants to make 3 batches of cookies, and only has 1/4 cup of sugar left in the pantry, how much additional sugar will the baker need to buy?" is an example of a word problem for the given equation.

To solve this word problem, we can use the equation 4/5 x 1/4 + 3/5 = to find out how much sugar is needed for one batch of cookies, and then multiply that amount by 3 to get the total amount of sugar needed for 3 batches.

The first part of the equation, 4/5 x 1/4, represents the amount of sugar needed for one batch of cookies, which is 1/5 cup. Adding the remaining 3/5 cup of sugar needed for the recipe gives a total of 4/5 cup of sugar per batch.

To find out how much additional sugar the baker needs to buy, we can multiply 4/5 by 3 (the number of batches), and then subtract the amount of sugar already in the pantry (1/4 cup). This gives us:

4/5 x 3 - 1/4 = 12/5 - 1/4 = 43/20

Therefore, the baker will need to buy 43/20 cups of additional sugar to make 3 batches of chocolate chip cookies.

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The company will have a Loss of $194,060

The lady paid premiums for 6 years, which amounts to a total premium of$ 5,940($ 990 * 6).

Still, the insurance company will pay the face value of the policy, which is   $ 00, If she dies.

Thus, the company's total payout would be $200,000, while their total income would be $ 5,940 in premiums.

The loss for the company would be the difference between the payout and the income

200,000-$ 5,940 = $ 194,060

Thus, the insurance company's loss in this scenario would be $194,060.

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The surface area of the right rectangular prism is 880 cm².

Given Top and bottom measurements of the rectangular prism = 15 cm and 11 cm

The front and back measurements of the rectangular prism = 11cm and 11cm

The two sides measurements of the rectangular prism = 11 cm and 14 cm

To find the surface area of the rectangular prism, we have to substitute the above values in the below equation,

Surface area = (top*bottom) + (front*back) + (sides)

Surface area = (15 cm * 11 cm) + (11 cm * 11 cm) + (11 cm * 14 cm)

Surface area = 330 cm² + 242 cm²  + 308 cm²  

Surface area = 880cm²

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Answer:

A dilation always produces a similar figure. Similar figures have the same shape but different sizes.

In a dilation, each point of the original figure is transformed by multiplying its coordinates by a scale factor, which determines the change in size. However, the shape and proportions of the figure remain unchanged. Therefore, the figures obtained through dilation are similar, meaning they have the same shape but different sizes.

The value of ║5p-3q║=180°.

It is given that line l₁ & l₂ are parallel and l₂ & l₃ are transversal then

they must follow the property that the alternate exterior angles must be equal i.e. ∠ q = 135°.

Also ∠ p + ∠ q = 180°.

Therefore, ∠ p = 45°.

Now to solve ║5p-3q║, substituting the values of p & q in the given equation

║5*45 - 3*135║ = 180°.

Hence, ║5p-3q║=180°.

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The magnitude of vector v is 4 and the direction is 125 degrees.

Given that vector u has a magnitude of 2 and a direction of 55 degrees, we can determine the magnitude and direction of vector v.

To find the magnitude of vector v, we can use the equation:

|v| = |-2u|

Since u has a magnitude of 2, we can substitute it into the equation:

|v| = |-2 * 2|

|v| = |-4|

|v| = 4

The magnitude of vector v is 4.

To find the direction of vector v, we can note that multiplying a vector by -1 (in this case, multiplying u by -2) reverses its direction. So the direction of v is the exact opposite of the direction of u.

Since the direction of u is 55 degrees, the direction of v is 55 degrees in the opposite direction. In the interval of 0 degrees ≤ θ < 360 degrees, the direction of v can be expressed as:

θ = 180 - 55

θ = 125 degrees

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No, there is not convincing evidence that p, the true proportion of 17-year-olds at this high school with driver's licenses, is not 0.75.

This is because the P-value of 0.9315 is very large, and the 95% confidence interval contains 0.75 (0.6300, 0.8811). This means that there is not enough evidence to reject the null hypothesis that the true proportion of 17-year olds with driver's licenses is 0.75. The 95% confidence interval also supports this, as it includes 0.75. Therefore, there is no convincing evidence to suggest that the student's claim is incorrect.

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An equivalent function of the given function in the form G(t) = [tex]ab^t[/tex] is G(t) = 24000[tex](0.9025)^t[/tex].

The given function for the value of a car that depreciates over time is G(t) = 24000[tex](0.95)^{2t[/tex]. To write an equivalent function of the form G(t) = [tex]ab^t[/tex], we need to rewrite the function using exponent rules.

First, we can rewrite [tex](0.95)^{2t[/tex] as [tex][(0.95)^{2}]^{t[/tex], which simplifies to [tex](0.9025)^t[/tex]. Therefore, we have:

G(t) = 24000[tex](0.9025)^t[/tex]

Next, we need to express 24000 as a product of two factors, a and b. We can choose a = 24000 and b = 0.9025, so we have:

G(t) = 24000[tex](0.9025)^t[/tex] = 24000[tex](0.9025)^t[/tex]

This function gives the value of the car at time t, where t is the number of years after the initial purchase, with a starting value of 24000 and a depreciation rate of 9.75% per year.

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Answer:

u = 2

Step-by-step explanation:

Alfonso received a 30% overall discount on the refrigerator and the stove together if the sale price of the refrigerator was 2/3 of the sale price of the stove. Therefore, statement (c) is the correct answer.

To see why, let's use some algebra. Let R be the original price of the refrigerator and S be the original price of the stove. Then the sale price of the refrigerator is 0.6R and the sale price of the stove is 0.8S. The total amount Alfonso paid is 0.6R + 0.8S.

To receive a 30% overall discount, Alfonso must have paid only 0.7 times the total original price, which is 0.7(R + S). Therefore, we have the equation:
0.6R + 0.8S = 0.7(R + S)

Simplifying this equation gives:
0.1R = 0.1S
R = S

So the original prices of the refrigerator and the stove were the same. This means that statement (C) is  true.

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Complete Question:  

Alfonso went to famous Sam's Appliances store and purchased a refrigerator and a stove. The sale price of the refrigerator was 40% off the original price and the sale price of the stove was 20% off the original price.

Which statement must be true to conclude that Alfonso received a 30% overall discount on the refrigerator and the stove together?

a) The sale prices of the refrigerator and the stove were the same.

b) The original prices of the refrigerator and the stove were the same.

c) The sale price of the refrigerator was twice the sale price of the stove.

d) The original price of the refrigerator was twice the original price of the stove.

The surface area of the triangular prism is 4302 mm²

A prism is a solid shape that is bound on all its sides by plane faces. The surface area of a prism is expressed as;

SA = 2B + ph where B is the base area and h is the height of the prism and p is the perimeter of the base.

Base area = 1/2 bh

= 1/2 × 36 × 24

= 36 × 12

= 432 mm²

Perimeter of the base = 36 + 30+30 = 96 mm²

SA = 2 × 432 + 96 × 36

SA = 846 + 3456

SA = 4302 mm²

Therefore the surface area of the triangular prism is 4302 mm².

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Kareem drove the truck for 150 miles.

As areem rented a truck for one day. there was a base fee of $19.99 , and there was an additional charge of 83 cents for each mile driven we will let the number of miles driven by Kareem be represented by 'x'. The total cost, including the base fee, can be represented as:

Total cost = Base fee + Additional charge per mile * Number of miles driven

$135.36 = $19.99 + $0.83x

Subtracting $19.99 from both sides and then dividing by $0.83, we get:

x = (135.36 - 19.99) / 0.83

x = 150

Therefore, Kareem drove the truck for 150 miles.

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A cube is a three dimensional figure with six square faces that are all congruent, meaning they are identical in shape and size.

Each face of the cube is perpendicular to the adjacent faces, forming right angles where they meet. The cube is a highly symmetrical shape, with all edges and angles equal in length and measure respectively. Its regularity and symmetry make it a popular shape in mathematics and engineering, often used as a model for buildings, containers, and other structures.

The cube's unique properties also make it an ideal shape for games, puzzles, and art, showcasing its versatility and significance in various fields.

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The mean value Taylor would use is 10.23 seconds.

To calculate the mean value for Taylor's recorded times, we add up the individual times (10 seconds, 10.3 seconds, and 10.4 seconds) to obtain a total of 30.7 seconds.

we divide this total by the number of trials, which in this case is 3.

30.7 seconds divided by 3 equals approximately 10.23 seconds.

The mean value of Taylor's recorded times for her car's trials is approximately 10.23 seconds.

The mean value is often used as a measure of central tendency to represent the average of a set of values.

In this case, it represents the average time recorded by Taylor during her trials.

By calculating the mean, we can compare this value with the mean times of other cars to assess performance.

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The simplification of the given algebraic expression is: -3x - 9y

Algebraic expressions are defined as the idea of expressing numbers with the aid of letters or alphabets without really specifying their actual values.

Now, the algebraic operations are known by the acronym PEMDAS which denotes:

P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction.

We want to find the sum of the algebraic expressions given as (-10x - 4y) and (7x - 5y) in simplest form.

Thus, we have:

(-10x - 4y) + (7x - 5y)

= -10x - 4y + 7x - 5y

= -3x - 9y

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Complete question:

Find an expression which represents the sum of (-10x - 4y) and (7x - 5y) in simplest terms.

The solution to the expression -(-4) (-6)-3/5(10+5)/1/3 is 15.

To solve this expression, we need to follow the order of operations, which is also known as PEMDAS (parentheses, exponents, multiplication and division, addition and subtraction):

Start with the parentheses: (-4) (-6)= 24, so we can rewrite the expression as:

24-3/5(15)/(1/3)

Now we need to simplify the expression inside the parentheses by adding 10 and 5, which equals 15. Then we need to simplify the division by multiplying the numerator by the reciprocal of the denominator, which is the same as dividing by 1/3. We can rewrite this expression as:

24-3/5*15*3

Multiply 3/5 and 15 first, then multiply that result by 3: 24-9.

Finally, subtract 9 from 24 to get the final answer: 15.

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The greatest possible width the land strip, in miles with the amount of material that has is 1/250 miles wide.

A quadrilateral with parallel sides that are equal to one another and four equal vertices is known as a rectangle. It is also known as an equiangular quadrilateral for this reason.

Rectangles can also be referred to as parallelograms since their opposite sides are equal and parallel.

A quadrilateral with equal angles and parallel opposing sides is referred to as a rectangle. Around us, there are a lot of rectangle items. The length and breadth of each rectangle serve as its two distinguishing attributes. The width and length of a rectangle, respectively, are its longer and shorter sides.

Let's say that her landing strip is x miles long, then its area would be:

1/6.x

We also know how big it is:

so,

1/6.x = 1/1500

x = 6/1500

x = 3/750

x = 1/250 miles

Therefore, possible width of the landing strip is 1/250 miles.

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Complete question:

Michelle is building a rectangular landing strip for airplanes .She has material to cover 1/1,500 of a square mile. The landing strip must be 1/6 of a mile long. With the amount of material that has , what is the greatest possible width the land strip, in miles?

Work Shown:

First we'll need the derivative of h(x)

[tex]h(\text{x}) = \sqrt{\text{x}}\\\\h(\text{x}) = \text{x}^{1/2}\\\\h'(\text{x}) = (1/2)\text{x}^{-1/2}\\\\h'(\text{x}) = \frac{1}{2\text{x}^{1/2}}\\\\h'(\text{x}) = \frac{1}{2\sqrt{\text{x}}}\\\\[/tex]

Then let f(x) = g(x)*h(x)

Use the product rule to evaluate f ' (9).

[tex]f(\text{x}) = g(\text{x})*h(\text{x})\\\\f'(\text{x}) = \frac{d}{d\text{x}}\left[g(\text{x})*h(\text{x})\right]\\\\f'(\text{x}) = g'(\text{x})*h(\text{x}) + g(\text{x})*h'(\text{x})\\\\f'(\text{x}) = g'(\text{x})*\sqrt{\text{x}} + g(\text{x})*\frac{1}{2\sqrt{\text{x}}}\\\\f'(9) = g'(9)*\sqrt{9} + g(9)*\frac{1}{2\sqrt{9}}\\\\f'(9) = 2*\sqrt{9} + 0*\frac{1}{2\sqrt{9}}\\\\f'(9) = 2*3 + 0\\\\f'(9) = 6\\\\[/tex]

The scatterplot  that would describe is Option A.

A scatter plot is described as  a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data.  

The slope - intercept form of the equation of a line is:

y = mx + c

where m =  the slope

c = the y-intercept

Only in the first scatterplot can the line of best fit intersect the y-axis at 2 if a line of best fit is drawn on each of the scatterplots. Only when a line of best fit is established on the first scatterplot is a slope of 1/2 conceivable.

That is, c = 2

m = 1/2

In conclusion, only the first scatterplot would have the line of best fit represented by the equation y = 1/2 x + 2.

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The value of the addition of the functions:

F(x) + g(x) = (8x² + 34x - 56)/(x²-3x-40).

To add the two rational functions F(x) and g(x), we first need to find a common denominator. In this case, the common denominator is (x+5)(x-8), since both denominators can be factored in this way.

F(x) needs to be multiplied by (x-8) on the top and bottom to get a common denominator of (x+5)(x-8), and g(x) needs to be multiplied by (x+5) on the top and bottom to get the same common denominator.

So, we have:

F(x) = (x+7)/(x+5) * (x-8)/(x-8) = (x² - x - 56)/(x² - 3x - 40)

g(x) = 7x/(x²-3x-40) * (x+5)/(x+5) = 7x(x+5)/(x+5)(x-8) = 7x(x+5)/(x²-3x-40)

Now that both functions have the same denominator, we can add them together:

F(x) + g(x) = (x² - x - 56)/(x² - 3x - 40) + 7x(x+5)/(x²-3x-40)

To simplify this expression, we need to combine the two fractions over the common denominator:

F(x) + g(x) = (x² - x - 56 + 7x² + 35x)/(x²-3x-40)

Combining like terms in the numerator:

F(x) + g(x) = (8x² + 34x - 56)/(x²-3x-40)

So, F(x) + g(x) = (8x² + 34x - 56)/(x²-3x-40).

To solve a rational function, we generally follow these steps:

In the case of F(x) and g(x), we already simplified the sum of the functions. We can see that the denominator factors as (x+5)(x-8), which means that the function is undefined at x = -5 and x = 8. These are vertical asymptotes.

To find any horizontal asymptotes, we can use the fact that the degree of the numerator is greater than or equal to the degree of the denominator. This means that there is no horizontal asymptote; instead, the function approaches infinity as x approaches infinity or negative infinity.

Finally, we can graph the function using this information and any other relevant points, such as intercepts.

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The value of h in the triangle shown above is calculated using proportion as: h = 4.

The two triangles shown are similar to each other based on the Angle-angle (AA) Similarity theorem. This implies that the length of their corresponding pair of sides would be proportional to each other.

Therefore, we have:

8/18 = h/9 [proportional sides of similar triangles]

Cross multiply:

h * 18 = 8 * 9

18h = 72

Divide both sides by 18:

18h/18 = 72/18 [Division property of equality]

h = 4

Therefore, the length of h in the given image is determined as: 4 units.

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the series converges for all x in the interval (-1/6, 1/6) in interval notation.

The given series is a geometric series with first term a=1 and common ratio r=6x. The series converges if and only if |r|<1.

So, |6x|<1

Solving this inequality, we get:

-1/6 < x < 1/6

To find all values of x for which the series converges, we need to analyze the given series:
∑(6x)^n

This is a geometric series with a common ratio of 6x. For a geometric series to converge, the absolute value of the common ratio must be less than 1:

|6x| < 1

Now, we can solve for the interval of x:

|-1/6| < |x| < |1/6|

Using interval notation, the range of values for which the series converges is:

(-1/6, 1/6)

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The Number of games lost cannot be negative .

The number of games the Senators won as "W." According to the given information, the Senators won 118 more games than they lost. This can be expressed as:

W = L + 118,

where "L" represents the number of games the Senators lost.

the Senators played a total of 78 games, which means the number of games they won and lost combined should equal 78:

W + L = 78.

Substituting the first equation into the second equation, we have:

(L + 118) + L = 78,

2L + 118 = 78,

2L = 78 - 118,

2L = -40,

L = -40/2,

L = -20.

Since the number of games lost cannot be negative.

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The loan with the smallest monthly payment is the 30-month loan at 6% annual simple interest rate, with a monthly payment of $45.83.

To determine the loan with the smallest monthly payment, we need to calculate the monthly payment for each loan option and compare them.

We can use the formula for monthly payment on a simple interest loan:

monthly payment = (principal + (principal * interest rate * time)) / total number of payments

where:

We can compute the monthly payments for each loan choice using this formula:

1. 12 Monthly interest rate = 0.0625/12 = 0.00521, monthly payment = (1250 + (1250 * 0.00521 * 12)) / 12 = $107.35

2. 18 months at 6.75%: monthly interest rate = 0.0675/12 = 0.00563, monthly payment = (1250 + (1250 * 0.00563 * 18)) / 18 = $81.96

3. 24 months at 6.5%: monthly interest rate = 0.065/12 = 0.00542, monthly payment = (1250 + (1250 * 0.00542 * 24)) / 24 = $66.14

4. 30 months at 6%: monthly interest rate = 0.06/12 = 0.005, monthly payment = (1250 + (1250 * 0.005 * 30)) / 30 = $45.83

Based on these calculations, the loan with the smallest monthly payment is the 30-month loan at 6% annual simple interest rate, with a monthly payment of $45.83.

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