MATHQDetermine what the selling price should be in order for the manufacturer’s weekly profit to be a maximum

Question

A clock manufacturer can produce a particular clock at a cost of $15 per clock. It is estimated that if the selling price of the clock is x dollars, then the number of clocks sold per week is 125 – x. Determine what the selling price should be in order for the manufacturer’s weekly profit to be a maximum.

Answer 1

P = (125-x) * (x-15)

P is profit, and it's # of clocks sold * the profit margin (price - cost).

P = (125-x) * (x-15) = -x^2 + 140x - 1875 (an inverted parabola).

The maximum value for an inverted parabola is the vertex.

If you use calculus, the vertex is where dp/dx = 0.
dP/dx = -2x + 140
0 = -2x + 140
2x = 140
x = $70 (selling price)
P = (125-x) * (x-15) = 55 * 55 = $3025

Check:
Check by finding the Profit for a $71 price and $69 price.
P = (125 - 71) * (71-15) = 56 * 54 = 3024
P = (125 - 69) * (69-15) = 56 * 54 = 3024 (check!)

Answer 2

70 $
then the no. of clocks sold is 55
which costs 825 $
selling price is 3850 $
which gives a profit of 3025 $