Applied Optimization-4.6 Calculus

Question

A store rents movies. If the rental fee is $2 ​each, the store knows they can rent 120 movies per week. For each additional​ $1 increase in the rental​ fee, the store loses 12 rentals per week. What rental fee should the store charge for a movie to maximize their​ revenue?

I have as far as

f(x)=(120-12x)(2+1x)

I think I need to use the product rule to find f'(x) and then solve for x. Then I will plug x into (2+1x)? I am struggling to find x.

Answer 1

That's a great set up for f(x). Now, you take f(x) and expand it out (use FOIL) and find the first derivative. Set that equal to zero (since you are looking for a max value, the tangent line is horizontal and has a slope of 0). Here is the solution: first derivative: 96-24x

set it equal to zero: 0=96-24x therefore, x=4

so, it looks like the price should be $6 after substituting x=4 into (2+x) portion of f(x)

Answer 2

You can just expand out the equation to get:

f(x) = 240 - 24x + 120x - 12 x^2

simplify:

f(x) = 240 + 96x -12 x^2.

Now take the derivative (to get an equation of the slope of the parabola for any given x), and set that derivative to zero (if the slope is 0, then you've reached a flat spot, either a minimum or a maximum).

f'(x) = 96 - 24 x

set it equal to 0:

96 - 24 x = 0

96 = 24x

4 = x

When x is 4, your parabola has a flat spot. You might notice that the sign of the x^2 term is negative, so as x gets very big or very small (negative), the two ends of the parabola go to negative infinity. So your flat spot is a maximum. Since the rental fee is $2 + x, the maximum revenue happens at a rental fee of

$2 + $4 = $6.