Question

A toy company has determined that the revenue generated by a particular toy is modeled by the following equation:
13x - 0.025x^2
The variable x is measured in thousands of toys produced, and r(x) is measured in thousands of dollars. What is the maximum revenue the company can earn with this toy?

Answer 1

Let's see the problem first:

1.) A toy company has determined that the revenue generated by a particular toy is modeled by the following equation: r(x)=13x - 0.025x²
The variable x is measured in thousands of toys produced, and r(x) is measured in thousands of dollars. What is the maximum revenue the company can earn with this toy?


The point at which a graph changes direction is called a maximum or a minimum.
How to find the maximum (or the minimum) of a function algebraically?

1st - Take the first derivative of the function:
r(x) = 13x - 0.025x²
r'(x) = 13 - 2 * ( 0.025) x
r'(x) = 13 - 0.05 x

2nd - Let the derivative equal zero and solve the equation this gives. There may be multiple solutions:
r'(x) = 13 - 0.05x
13 - 0.05x = 0

Adding 0.05x in both terms:
13 = 0.05x

Dividing both sides by 0.05:
13/0.05 = x

260 = x

Since the variable x is measured in thousands of toys produced, this represents 260,000 toys.

3rd - For every solution (only one in this case), substitute x back into the original function to find the equivalent y value:
r(x) = 13x - 0.025x²
r(260) = 13(260) - 0.025(260)²
r(260) = 1690

Since r(x) is measured in thousands of dollars, the maximum revenue the company can earn with this toy will be $ 1,690,000.

ANSWER → $ 1,690,000

------------------------

Answer 2

Medal and fan?! Thx