Question

The profit (in R1000s) yielded by a company, using a machine that produces bottle caps, is dependent on the average speed at which the machine runs. 

The profit (P) is calculated using the formula:

$$P=-3v^2+30v$$

where $v$ is the average speed (in kilometers per hour) and $v>0$

1. Calculate the average speed at which neither a profit , nor a loss is yielded.
2. Determine at what average speed the machine should run so that the maximum profit is obtained.
3. Hence, or otherwise, calculate the resulting maximum profit.

Answer 1

1. P =  -3v^2 + 30v
Neither profit nor loss at P = 0
 3v(v -10) = 0
v = 0 or v =10
v = 10 km/hr.

2. P =  -3v^2 + 30v

dP/dv =  -6v + 30 = 0

        v  = 5 km/hr

3. Pmax (in R1000) = 3(5)^2 + 30(5) =  75 (in R1000) or R75 000