Question

Your business sells cupcakes in boxes of 10. The demand equation is

x = −5p + 100
In the above formula x is the number of boxes you sell in one month for a unit price (per box) of p dollars. The cost of producing x boxes is
C = $50 + 6xSet up the profit function P in terms of an arbitrary number of boxes x alone.

Answer 1

`\displaystyle P=(250 -70x+x^2)/(5)`

Explanation:

Revenue, Cost and Profit

In basic economics, the study of these 3 variables provides some insights about the businesses' health. Given the demand equation and the total cost function, it's possible to build up some interesting outcomes and even to find the optimal levels of production to maximize profits

The demand equation is given as

`x = -5p + 100`

where x is the number of boxes sold in one month for a unit price (per box) of p dollars.

We also know that the cost of producing x boxes is

`C = 50 + 6x`

The revenue function can be easily computed if we know the price as a function of x. The price by the number of boxes sold will give us the revenue function. We need to isolate p in the demand equation

`x = -5p + 100`

`5p =100-x`

`p=\displaystyle (100-x)/(5)`

Now the revenue function is

`\displaystyle R=px=(100-x)/(5)x`

`\displaystyle R=(100x-x^2)/(5)`

The profit function P is always computed as the difference between the cost and the revenue

`\displaystyle P=C-R=50 + 6x-\left ( (100x-x^2)/(5) \right )`

`\displaystyle P=(250 + 30x-100x+x^2)/(5)`

`\displaystyle P=(250 -70x+x^2)/(5)`