1. In this problem, we 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 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 by following 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.
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 pieces of PVC plumbing pipe be?
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To solve this problem, we will go through each part step by step:
a. To find the demand equation in the form p = mx + b, we can use the two points given: (42, $20) and (52, $10).
Using the slope formula:
m = (p2 - p1) / (x2 - x1)
= ($10 - $20) / (52 - 42)
= -$10 / 10
= -$1
Using the point-slope formula:
p - p1 = m(x - x1), where (x1, p1) is one of the points.
p - $20 = -$1(x - 42)
p - $20 = -$x + $42
p = -$x + $62
So the demand equation is p = -$x + $62.
b. The revenue equation is given by R = xp, where x is the quantity of the product sold and p is the price per unit.
Substituting the demand equation from part a, we have
R = x(-$x + $62) = -$x^2 + $62x
So the revenue equation is R = -$x^2 + $62x.
c. In this context, the fixed cost is the b value mentioned in the demand equation. Since it represents a fixed cost, the fixed cost in this case is $62.
d. The cost equation can be found by adding the fixed cost ($300) and the variable cost ($6x, where x is the number of tile sets).
So, C = $6x + $300.
e. The profit equation is given by subtracting the cost equation (C) from the revenue equation (R):
P = R - C = (-$x^2 + $62x) - ($6x + $300)
= -$x^2 + $62x - $6x - $300
= -$x^2 + $56x - $300
So the profit equation is P = -$x^2 + $56x - $300.
f. To find the profit made from selling 20 tile sets per month, we substitute x = 20 into the profit equation:
P = -$(20^2) + $56(20) - $300 = -$400 + $1120 - $300
= $420
The profit made from selling 20 tile sets per month is $420.
g. To find the profit made from selling 25 tile sets per month, we substitute x = 25 into the profit equation:
P = -$(25^2) + $56(25) - $300 = -$625 + $1400 - $300
= $475
The profit made from selling 25 tile sets per month is $475.
h. To find the profit made from selling no tile sets each month, we substitute x = 0 into the profit equation:
P = -$(0^2) + $56(0) - $300 = -$300
The profit made from selling no tile sets each month is -$300. This means that the company would have a loss of $300 if they don't sell any tile sets.
i. To find the quantity of tile sets per month that yields the highest profit, we can find the x-value of the vertex of the profit equation.
The x-value of the vertex can be found using the formula:
x = -b / (2a), where a, b, and c are the coefficients in the quadratic equation.
In this case, a = -1, b = $56, and c = -$300.
So, x = -($56) / (2(-1)) = $28 / 2 = 14.
Therefore, selling 14 tile sets per month would yield the highest profit.
j. To find the profit earned from selling 14 tile sets per month, we substitute x = 14 into the profit equation:
P = -$(14^2) + $56(14) - $300 = -$196 + $784 - $300
= $288
The profit earned from selling 14 tile sets per month is $288.
k. To find the price at which the tile sets should be sold to realize this profit, we plug the value of x = 14 back into the demand equation from part a:
p = -$x + $62 = -$(14) + $62 = $-14 + $62 = $48
The tile sets should be sold at a price of $48 to realize a profit of $288.
2. To find the break-even values, we set the profit equation equal to zero (P = 0) and solve for x:
0 = -$(x^2) + $56x - $300
This equation can be solved by using factoring, completing the square, or the quadratic formula. Once we find the values of x, those will be the break-even points.
3. In scientific notation, Home Depot's 2002 sales of $58,200,000,000 would be written as 5.82 x 10^10 and 2006 sales of $90,800,000,000 would be written as 9.08 x 10^10.
a. To find the increase in sales from 2002 to 2006, we subtract the 2002 sales from the 2006 sales:
Increase = 9.08 x 10^10 - 5.82 x 10^10 = 3.26 x 10^10.
To find what percent the increase is of the 2002 sales, we divide the increase by the 2002 sales and multiply by 100:
Percent increase = (3.26 x 10^10 / 5.82 x 10^10) x 100 = 56%.
b. The percent growth in Home Depot's sales from 2002 to 2006 is 56%.