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(x) = 230 + 40x - 0.5x2 What expenditure for advertising will yield a maximum profit?

A) 40
B) 0.5
C) 230
D) 20

Answer 1

To maximize profit based on the given quadratic profit function, the advertising expenditure should be $4000, which corresponds to the vertex of the parabola formed by the function. So the correct option is A.

Step-by-step explanation:

The student is asking about finding the amount of advertising expenditure that will yield the maximum profit for a company based on the given quadratic profit function P(x) = 230 + 40x - 0.5x2. To find the advertising expenditure that maximizes profit, we need to identify the vertex of the parabola represented by this quadratic equation, which occurs at the value of x where the derivative of P(x) concerning x is zero.

To do this, we take the derivative of P(x) and set it equal to zero:

• P'(x) = 40 - x
• 0 = 40 - x
• x = 40

After finding the value of x, we see that when x equals 40 (in hundreds of dollars), the profit P(x) is at its maximum. This means the profit-maximizing level of advertising expenditure is $4000 (since x represents hundreds of dollars).

Answer 2

Option A

Step-by-step explanation:

Given that 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(x) = 230 + 40x - 0.5x^2`

We can use derivative test to find maximum

Differentiate two times the profit funciton

`P'(x) = 40-x\\P`

Equate I derivative to 0

we get

`x=40`

II derivative is alrways negative

Hence profit is maximum when x=40

Option A is right