# Algebra

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Sorry, it didn't paste the first time, I was wondering if I got these correct; especially number 2 at the bottom.

1. 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).

P=-x+62

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.

R = -x^2 + 62 x

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. 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.

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

b = 300 (fixed costs)

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

m = 6 (variable costs)
x = number of tile sets

C = 6x + 300

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.

Profit = Revenue - Costs
Profit = -x² + 62x - (6x + 300)
Profit = -x² + 62x - 6x - 300
Profit = -x² + 56x - 300

f.What is the profit made from selling 20 tile sets per month?
Profit = -x² + 56x - 300
Profit = -(20)² + 56(20) - 300
Profit = -400 + 1120 - 300
Profit = 420

g.What is the profit made from selling 25 tile sets each month?
Profit = -x² + 56x - 300
Profit = -(25)² + 56(25) - 300
Profit = -625 + 1400 - 300
Profit = 475

Profit = -x² + 56x - 300
Profit = -(0)² + 56(0) - 300
Profit = -0 + 0 - 300
Profit = -300
The company will spend \$300 for fixed costs regardless of sales, so without any sales, they lose \$300 from their account each month.

i.Use trial and error to find the quantity of tile sets per month that yields the highest profit.
For a quadratic equation with a negative leading coefficient, there is always a maximum value at the vertex, which is located on the equation's Axis of Symmetry. The Axis of Symmetry is found from x = -b/(2*a), so here you get:

Profit = -x² + 56x - 300
x = -56/(2*-1) = 28
28 tile sets would generate the maximum profit.

j.How much profit would you earn from the number you found in part i?
Profit = -x² + 56x - 300
Profit = -(28)² + 56(28) - 300
Profit = -784 + 1568 - 300
Profit = 484

k.What price would you sell the tile sets at to realize this profit (hint, use the demand equation from part a)?
P=-x+62
P= -28 + 62
Price = 34

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.
-x² + 56x - 300 = 0
-(x² - 56x + 300) = 0
-(x - 50)(x - 6) = 0
x = 50 or x = 6
The company will break even if they sell 50 sets or 3 sets of tiles.