a. Suppose that a market research company finds that at a price of p = $20, they would sell x = 62 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 = 72 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)).

First Point:

Second Point:

Find the slope:

m = =

Find the equation of the line:

Solve the equation in terms of p:

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.

Hint: This is an equation in terms of x

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?

Hint: This is just a simple number.

d. Find the cost equation for the tile. Write your answer in the form C = mx + b.

Hint: m = the variable costs for each set of tile.

b = the fixed cost (from part c)

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.

Hint: This is your answer to part (b) minus your answer to part (d). Do not forget to distribute the negative sign as you simplify.

f. What is the profit made from selling 10 tile sets per month?

Hint: Plug in 10 for x into your answer from part (e).

g. What is the profit made from selling 30 tile sets each month?

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

Hint: Plug in 0 for x into your answer from part (e).

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

Hint: No, there is not unlimited profit.

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

Hint: You should all ready have this number from your trial and error answers above.

k. What price would you sell the tile sets at to realize this profit?

Hint: Use the demand equation from part (a) and your answer from part (i).

2. The break even values for a profit model are the values for which you earn $0 in profit.

Question: Your Answer:

Using a Profit equation of P = -x2 + 25x - 100

solve for P = 0, and find the break even values.

Hint: Take your Profit equation given, set it equal to zero, and solve for x. You will need to factor out the negative first, then factor the remaining quadratic equation.

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

Questions: Your Answers:

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

2002 sales:

2006 sales:

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

(Subtract 2002 sales from 2006 sales)

• Find what percent that increase is of the 2002 sales.

Percent increase =

(source: Home Depot Annual Report for FY 2006: http://www6.homedepot.com/annualreport/index.html)

4. 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 PVC plumbing pipe indicated in the diagram be?

(Hint: Write an equation using two points in the form (x,p)).

First Point:

Second Point:

Find the slope:

m = =

Find the equation of the line:

Solve the equation in terms of p:

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.

Hint: This is an equation in terms of x

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?

Hint: This is just a simple number.

d. Find the cost equation for the tile. Write your answer in the form C = mx + b.

Hint: m = the variable costs for each set of tile.

b = the fixed cost (from part c)

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.

Hint: This is your answer to part (b) minus your answer to part (d). Do not forget to distribute the negative sign as you simplify.

f. What is the profit made from selling 10 tile sets per month?

Hint: Plug in 10 for x into your answer from part (e).

g. What is the profit made from selling 30 tile sets each month?

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

Hint: Plug in 0 for x into your answer from part (e).

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

Hint: No, there is not unlimited profit.

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

Hint: You should all ready have this number from your trial and error answers above.

k. What price would you sell the tile sets at to realize this profit?

Hint: Use the demand equation from part (a) and your answer from part (i).

2. The break even values for a profit model are the values for which you earn $0 in profit.

Question: Your Answer:

Using a Profit equation of P = -x2 + 25x - 100

solve for P = 0, and find the break even values.

Hint: Take your Profit equation given, set it equal to zero, and solve for x. You will need to factor out the negative first, then factor the remaining quadratic equation.

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

Questions: Your Answers:

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

2002 sales:

2006 sales:

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

(Subtract 2002 sales from 2006 sales)

• Find what percent that increase is of the 2002 sales.

Percent increase =

(source: Home Depot Annual Report for FY 2006: http://www6.homedepot.com/annualreport/index.html)

4. 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 PVC plumbing pipe indicated in the diagram be?

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