Quantitative techniques

Question

a) A manufacturing firm finds that the daily costs of producing x units of a product is given

by:

𝒄 = 𝟎. 𝟎𝟐𝟓𝒙²

+ 𝟏𝟑𝒙 + 𝟏𝟎𝟎

a. If each of the units is sold for Ksh 20, determine the minimum numbers that

must be produced and sold daily to ensure that there is no loss to the company.

(3 Marks)

b. If the selling price is increased by 45%, at what level of production would the

company not incur any losses? (2 marks)

Answer 1

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:

0.025x² + 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:

0.025x² - 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:

0.025x² + 13x + 100 = 29x

Simplify

0.025x² - 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