Julia S. answered • 07/28/21

Calculus Made Manageable

For part a:

In order for the company to break even every day, the cost function must equal the revenue function. Our cost function was given and the revenue function looks like:

R(x) = 20x

We can now set them equal to each other to get:

0.025x^{2} + 13x + 100 = 20x

To solve for where the function equals 0, we can move the 20x over and use quadratic formula to solve. Our simplified equation should look like:

0.025x^{2} - 7x + 100 = 0

Using quadratic formula, we get roots of 15.1 and 264.9. We need the minimum units sold, and since we can't sell 0.1 of a unit, we round up to 16 units.

**x = 16 units**

For part b:

If the selling price increases, we can predict it should take fewer units to reach the daily breakeven goal. Our new price is:

20 * 1.45 = $29 per unit

We can use the same equation as before with our new unit price - that is:

0.025x^{2 }+ 13x + 100 = 29x

Simplify

0.025x^{2} - 16x + 100 = 0

Using quadratic formula we get zeros at 6.3 and 633.7. We take the minimum of 6.3 units sold and round up to 7 units.

**x = 7 units**