Question

The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) the company spends on advertising according to the model P = 200 + 50x − 0.5x2. What expenditure for advertising yields a maximum profit?

Answer 1

Answer:

50 (hundred dollars)

Explanation:

To get maximum profit, we differentiate P and the result, P', we set equal to zero.

$P(x)=200+50x-0.5x^{2}\\ \\P'(x)=50-x\\\\0=50-x\\\\x=50$

Every time you need to optimize a function, you differentiate and equal to zero to get the answer or answers!

In this case if we plug 50 into P we get:

$P(x)=200+50x-0.5x^{2}\\\\P(50)=200+50(50)-0.5(50)^{2}\\\\P(50)=200+2500-1250\\\\P(50)=1450$

Remember that the function yields results in hundred of dollars, so the profit would be equal to 145000 dollars.

Look at the graph of the P function, at x= 50 we see the vertex of the parabola which is the highest point over the entire graph, which is also its maximum profit or value.