If the equation for revenue is:

R = xp

and the equation from a is:

p = -x + 62

And you substitute (-x + 62) for p in the revenue equation, it should be:

R = x(-x + 62) = -x^2 + 62x

You can check it by using real numbers. Let's say you sold 52 tile sets.

R = -(52^2) + (62 * 52)

R = -2704 + 3224 = $520

Now use the same numbers from your demand curve from a. If you sell 52 sets, you know the price (p) is $10 (the point (10,52) was given to you in question a). So if you sell 52 sets at $10 each, you get $520. That matches the revenue you calculated using the Revenue equation.

c. Fixed cost is $300

d. Cost equation: C = mx + b where b is fixed cost of $300, m is the cost per tile set ($6) and x is the number of sets sold. The equation you have is correct:

C = 6x + 300

e. The key is this sentence:

The profit made from the sale of tiles is found by subtracting the costs from the revenue.

So the equation for profit (P) should be:

P = R - C

In terms of x:

P = (-x^2 + 62x) - (6x + 300)

= -x^2 + 62x - 6x - 300

= -x^2 + 56x - 300

f. Profit from selling 20 tile sets:

P = -(20^2) + (56 * 20) - 300

= -400 + 1120 - 300

= $420

You can check it this way. We know that each tile set costs $6. So the cost of 20 tile sets (plus the fixed costs) is:

C = 6x + 300

= (6 * 20) + 300

= 120 + 300

= 420

How much are we selling the tile sets for if we sell 20 in a month? Use the demand equation from question a:

p = -x + 62

= -20 + 62

= 42

That is, if the price of the tile sets is $42, we can expect to sell 20 sets in a month.

Revenue for 20 sets is:

R = xp = 20 * 42 = $840

Profit = Revenue - Cost = $840 - $420 = $420

g. Profit from selling 25 tile sets:

P = -(25^2) + (56 * 25) - 300

= -625 + 1400 - 300

= $475

You can check it this way. We know that each tile set costs $6. So the cost of 25 tile sets (plus the fixed costs) is:

C = 6x + 300

= (6 * 25) + 300

= 150 + 300

= 450

How much are we selling the tile sets for if we sell 25 in a month? Use the demand equation from question a:

p = -x + 62

= -25 + 62

= 37

That is, if the price of the tile sets is $37, we can expect to sell 25 sets in a month (we sell more because it's cheaper).

Revenue for 25 sets is:

R = xp = 25 * 37 = $925

Profit = Revenue - Cost = $925 - $450 = $475

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

P = -x^2 + 56x - 300

= 0 + 0 - 300

= -$300

We have a loss of $300. That is because there is no revenue and we still have to pay for the fixed costs (rent, insurance, payroll, etc).

If you know how to use a spreadsheet, then the next series of questions is much easier to figure out. You just fill one column with prices from $0 to $62, input the Profit formula into an adjacent cell, and you should be able to figure out the answers to all the next questions (if you want me to send you the spreadsheet, let me know).

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

For the trial and error method, if you don't want to use a spreadsheet, try picking several quantities (x) and see how much profit you generate. Remember, the quantity you sell per month is going to be related to the price you set. If your price is too expensive, people aren't going to buy as many (x goes down), and if your price is too cheap, you won't make enough on each set of tiles to cover your costs. You know that at 62 dollars, no one is going to buy it (according to the demand curve in question a), so somewhere between $0 and $62 is an ideal price where you realize the best profit. Try going halfway (with a price of $31, then x will be $62 - 31 = 31), and then go one to each side (30 and 32). Figure out which way you need to go (up or down) in quantity to get the biggest profit. That is, if x = 30 gives you more profit than x = 31, then try x = 29, and so on until the profit starts going down again.

Looking at my spreadsheet, I see the biggest profit when quantity (x) = 28.

j. When x = 28, profit P = $484

k. Price would be p = -x + 62 = -28 + 62 = $34

2. There are actually 2 break-even values: x = 6 and x = 50. That is, if you sell exactly 6 sets, you don't make a profit, but you also don't make a loss. If you sell less than 6, you get a loss, if you sell more than 6 but less than 50, you get a profit. If you sell 50, again, you don't see a profit or a loss. However, if you start selling more than 50, you get a loss. Why? Because the price that you have to set in order to sell more than 50 sets is so low that your profit would be minimal. For example, in order to sell 55 sets in a month, you would have to set the price at $7. Since the cost of each set is $6, you would only be making $1 on each set, which isn't enough to cover your fixed costs of $300.