SOLUTION: 3) You open a hat stand in the mall with an initial start-up cost of $1500 plus 50 cents for every hat you stock your booth with. You decide to charge $5 for each hat. b) What

Question 1132801: 3) You open a hat stand in the mall with an initial start-up cost of $1500 plus 50 cents for every hat you stock your booth with.
You decide to charge $5 for each hat.
b) What is your revenue function?
c) What is your profit function?
d) How much money will you make if you sell 1000 hats?
e) How many hats do you need to sell in order to break even?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
your cost equation is 1500 + .5 * x

your revenue equation is 5 * x.

revenue minus cost = profit.

your profit equation is 5 * x - (1500 + .5 * x)

simplify this to get 5 * x - 1500 - .5 * x

combine like terms to get 4.5 * x - 1500.

your profit equation becomes 4.5 * x - 1500.

if you stock and sell 1000 hats, .....

your cost will be 1500 + .5 * 1000 = 2000.

your revenue will be 5 * 1000 = 5000.

your profit will be 4.5 * 1000 - 1500 = 3000

revenue minus cost = profit.

this results in 5000 - 2000 = 3000.

the formulas check out ok.

your solutions are:

b) What is your revenue function?

your revenue function is r = 5 * x.

c) What is your profit function?

your profit function is p = 4.5 * x - 1500.

d) How much money will you make if you sell 1000 hats?

if you stock and sell 1000 hats, you will make 5000 in revenue with a profit of 3000.

e) How many hats do you need to sell in order to break even?

you will break even when your revenue equals your cost.

alternatively, you will break even when your profit is equal to 0.

using the revenue and cost equations, set revenue equals cost to get:

r = c becomes 5*x = 1500 + .5 * x
subtract .5 * x from both sides of this equation to get 5*x - .5 * x = 1500
combine like terms to get 4.5 * x = 1500
divide both sides by 4.5 to get x = 333.33333..... = 333 and 1/3 hats.

using the profit equation, set profit equal to 0 to get 4.5 * x - 1500 = 0
add 1500 to both sides to get 4.5 * x = 1500.
divide both sides by 4.5 to get x = 333 and 1/3 hats.

since you can't sell 1/3 of a hat, you either lose a little when you sell 333 or you gain a little when you sell 334.

at 333 hats, your profit is 4.5 * 333 - 1500 = -1.5 which is the same as a loss of 1.5.

at 334 hats, your profit is 4.5 * 334 - 1500 = 3.

using the revenue and cost equtions, .....

at 333 hats, your revenue is 5 * 333 = 1665 and your cost is 1500 + .5 * 333 = 1666.5 for a profit of 1665 - 1666.5 = -1.5 which is the same as a loss of 1.5.

at 334 hats, your revenue is 5 * 334 = 1670 and your cost is 1500 + .5 * 334 = 1667 for a profit of 1670 - 1667 = 3.

break even is technically at 333 and 1/3 hats even though that is not practical.

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