The revenue, R(x), of producing and selling x Awesome Hearing Aids is modeled by the function R(x) = -2 x^2 + 103 x

Question

The revenue, R(x), of producing and selling x Awesome Hearing Aids is modeled by the function
R(x) = -2 x^2 + 103 x

How many hearing aids need to be produced and sold in order to maximize the revenue? (Round to the nearest whole number.)

The revenue is at a maximum after _____hearing aids are produced and sold.

What is the revenue? (Round answer to the nearest cent.)

The maximum revenue is $______

Answer 1

Revenue is a function, just like everything else that you've seen so far in calculus. How do we find the maximum of a function? We take the derivative and set it equal to zero. Why? because at a max or min, the slope of the function is zero.

 

R(x) = -2 x² + 103x

R'(x) = -4x + 103

 

Set that equal to zero...

 

-4x +103 = 0

103 = 4x

103/4 = x

x = 25.75 ~ 26, as the problem asked us to round to the nearest whole number.

 

Thus, they need to sell 26 hearing aids.

 

The maximum revenue is found by plugging in this value for x into the original revenue function.

 

R(28) = -2 * (25.75)² + 103 * 25.75

         = 1326.125

         ~ 1236.16 rounded to the nearest cent

 

I hope this helps! Please let me know if you have any questions about this, and I will get back to you as soon as I can.