Suppose a baby food company has determined that its total revenue
R
for its food is given by

R= -x^3 +93x^2+1680x
where
R
is measured in dollars and
x
is the number of units (in thousands) produced. What production level will yield a maximum revenue?

A production level of
thousand units will maximize revenue.

Question

Suppose a baby food company has determined that its total revenue 
R
 for its food is given by

R= -x^3 +93x^2+1680x
where 
R
 is measured in dollars and 
x
 is the number of units (in thousands) produced. What production level will yield a maximum revenue?

A production level of 
 thousand units will maximize revenue.

BankrolHayden · Answer

a lot i'd say like over a thousand

SmokeyBrown · Answer

The way I tackled this problem is by taking the derivative of the original equation.
This is a calculus technique that basically gives us the rate of change of the equation at any given point along its curve. This can help us because, at the maximum of the equation, the rate of change will be 0.
So, we can get the derivative of each part of the equation by taking the exponent, multiplying the part by the exponent, and subtracting 1 from the exponent. 
For example, -x^3 becomes -3x^2. Likewise, 93x^2 becomes 186x, and 1680x becomes 1680.
Our derivative equation is [-x^3 + 186x +1680]

SmokeyBrown · Answer

Now that we have the derivative equation, we can find what value of x makes the derivative equal to 0. This is easy if we complete the square. We separate the parts with "x" on one side of the equation, and also make sure that the part with the largest exponent has no coefficient. The result of this is:
x^2 - 62x = 560
We can do some factoring to complete the square, and the result of that is:
(x-31)^2 = 1521
Taking the square root of both sides leaves us with
x-31 = +-39 (positive or negative because of the square root, but we're only interested in positive answers for this question
And finally, x = 70
Going back to the original equation, this means that the maximum should occur when x = 70

SmokeyBrown · Answer

We can even verify this result using a graphing calculator... It seems like our method turned out the correct answer. I used desmos.com for this one "But couldn't we have used a graphing calculator from the beginning?" Absolutely! If we did that, then we'd be helpless when it comes to solving these kinds of problems without a calculator. That's why it's best to try finding the answer first and then using a calculator to check your work

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