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# Math Problems: Application Practice

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Application Practice: Answer the following questions to the problem.

1. In this problem, we will analyze the profit found for sales of decorative tiles. A demand equation (sometimes called a demand curve) shows how much money people would pay for a product depending on how much of that product is available on the open market. Often, the demand equation is found empirically (through experiment, or market research).

a. Suppose that a market research company finds that at a price of p = \$20, they would sell x = 42 tiles each month. If they lower the price to p = \$10, then more people would purchase the tile, and they can expect to sell x = 52 tiles in a month's time. Find the equation of the line for the demand equation. Write your answer in the form p = mx + b. (Hint: Write an equation using two points in the form (x,p)). A company's revenue is the amount of money that comes in from sales, before business costs are subtracted. For a single product, you can find the revenue by multiplying the quantity of the product sold, x, by the demand equation, p.

b. Substitute the result you found from part a into the equation R = xp to find the revenue equation. Provide your answer in simplified form. The costs of doing business for a company can be found by adding fixed costs, such as rent, insurance, and wages, and variable costs, which are the costs to purchase the product you are selling. The portion of the company's fixed costs allotted to this product is \$300, and the supplier's cost for a set of tile is \$6 each. Let x represent the number of tile sets.

c. If b represents a fixed cost, what value would represent b?

d. Find the cost equation for the tile. Write your answer in the form C = mx + b. The profit made from the sale of tiles is found by subtracting the costs from the revenue.

e. Find the Profit Equation by substituting your equations for R and C in the equation. Simplify the equation.

f. What is the profit made from selling 20 tile sets per month?

g. What is the profit made from selling 25 tile sets each month?

h. What is the profit made from selling no tile sets each month? Interpret your answer.

i. Use trial and error to find the quantity of tile sets per month that yields the highest profit.

j. How much profit would you earn from the number you found in part i?

k. What price would you sell the tile sets at to realize this profit (hint, use the demand equation from part a)?

2. The break even values for a profit model are the values for which you earn \$0 in profit. Use the equation you created in question one to solve P = 0, and find your break even values.

3. In 2002, Home Depot's sales amounted to \$58,200,000,000. In 2006, its sales were \$90,800,000,000.

a. Write Home Depot's 2002 sales and 2006 sales in scientific notation.

You can find the percent of growth in Home Depot's sales from 2002 to 2006, follow these steps:
- Find the increase in sales from 2002 to 2006.
- Find what percent that increase is of the 2002 sales.

b. What was the percent growth in Home Depot's sales from 2002 to 2006? Do all your work by using scientific notation.

(source: Home Depot Annual Report for FY 2006: http://www6.homedepot.com/annualreport/index.html
1. A customer wants to make a teepee in his backyard for his children. He plans to use lengths of PVC plumbing pipe for the supports on the teepee, and he wants the teepee to be 12 feet across and 8 feet tall (see figure). How long should the pieces of PVC plumbing pipe be?

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## Geometry Problems

Geometry has many practical applications in everyday life. Estimating heights of objects, finding distances, and calculating areas and volumes are commonplace. One of the most fundamental theorems in geometry, the Pythagorean Theorem, allows us to make many of these calculations. The Pythagorean Theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, as shown in the diagram below.

The problems in this Unit will give you an opportunity to practice these applications.
Solve the following problems and submit them in a Word document.
1. A Little League team is building a backstop for its practice field. It is made up of two right angles as shown below. The backstop extends 24 feet 8 inches out in each direction and the center pole is 6.5 yards high. All sides of the backstop including base and the center pole are to be made of aluminum tubing. How many feet of tubing should the team buy? How many square feet of the backstop must be covered by a screen?

2. An Indian sand painter begins his picture with a circle of dark sand. He then inscribes a square with a side length of 1 foot inside the circle. What is the area of the circle?

3. Three buildings abut as shown in the diagram below. What are the dimensions of the courtyard and what is the perimeter of the building?

4. A cylindrical can is just big enough to hold three tennis balls. The radius of a tennis ball is 5 cm. What is the volume of air that surrounds the tennis balls?

See attached file for full problem description.

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