Question

A small business makes cookies and sells them at the farmer's market. The fixed monthly cost for use of a Health Department–approved kitchen and rental space at the farmer's market is $790. The cost of labor, taxes, and ingredients for the cookies amounts to $0.24 per cookie, and the cookies sell for $6.00 per dozen. (See Example 7) Write a linear cost function representing the cost C(x) to produce x dozen cookies per month. Write a linear revenue function representing the revenue R(x) for selling x dozen cookies. Write a linear profit function representing the profit for producing and selling x dozen cookies in a month. Determine the number of cookies (in dozens) that must be produced and sold for a monthly profit. If 150 dozen cookies are sold in a given month, how much money will the business make or lose?

Answer 1

Answer:

Cost function:

$C(x)=790+2.88x$

Revenue function

$R(x)=6x$

Profit function

$P(x)=3.12x-790$

Dozens needed for a specific profit P

$x=\dfrac{P-790}{3.12}$

If 150 dozen cookies are sold the profit is negative, so the business is loosing $322.

Step-by-step explanation:

We can list the costs as:

- Fixed monthly cost, $790/month.

- Variable costs, $0.24/cookie, which are (12*0.24) = $2.88 a dozen.

Then, we can write the cost function C(x) as:

$C(x)=790+2.88x$

being C(x): the monthly cost and x: the number of dozens produced per month.

The revenue can be calculated as the price ($6 a dozen) multiplied by the number of dozens x:

$R(x)=6x$

The profit can be calculated substracting the total cost C(x) from the revenue R(x).

$P(x)=R(x)-C(x)\\\\P(x)=(6x)-(790+2.88x)=(6-2.88)x-790\\\\P(x)=3.12x-790$

The number of cookies (in dozens) that must be produced and sold for a monthly profit can be calculated from the previous equation for P(x).

For a monthly profit P, the number of dozens that need to be sold are:

$P=3.12x-790\\\\P+790=3.12x\\\\\\x=\dfrac{P-790}{3.12}$

If 150 dozen cookies are sold, the profit made is:

$P(150)=3.12(150)-790=468-790=-322$

The profit is negative, so the business is loosing $322.