I was wondering if I had number 2 correct; the only way to know the answer was to provide the entire answer for number 1. Thanks in advance.

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

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

It looks correct to me, including the part #2 that you asked about. You also explained the problem and your work very clearly. I appreciate that. Nice job

yeahoooo ty ty ty =)

To find the break-even values for the profit model, you need to solve the equation for P = 0. In this case, the profit equation is given as:

Profit = -x² + 56x - 300

To solve for P = 0, set the profit equation equal to zero:

-x² + 56x - 300 = 0

Next, rearrange the equation to make it easier to solve:

-(x² - 56x + 300) = 0

Now, you need to factorize the quadratic equation. However, if it can't be factored easily, you can try using the quadratic formula. In this case, the equation can be factored by trial and error as:

-(x - 50)(x - 6) = 0

So the two possible break-even values are x = 50 and x = 6.

This means that the company will break even if they sell either 50 sets or 6 sets of tiles.