Calculus word problem

Question

A salsa company estimates that their profit function (in dollars) is given by P(x) = -2x^3 + 840sqrt x - 470, where x is the quantity produced, in hundreds of cases. At what rate is their profit increasing or decreasing when they produce 900 cases of salsa? Make sure your answer is complete.

Answer 1

Our profit function is P(x)= -2x³+840x^1/2-470. I rewrote this with a fractional exponent, because that makes it easier to find the derivative. If you ever see a square root or strange radical, try to rewrite it as a fractional exponent :)

The derivative of P(x) will give you the rate at which the profit is changing, so we need to find the value of the derivative when we have 900 cases (be careful here! You might be tempted to say x=900, but they said x is the number of hundred cases. That would mean x=9.)

P'(x) = -3*2x² + (1/2)840x^-1/2

Multiplying and simplifying, we get:

P'(x) = -6x² + 420x^-1/2

Now we will evaluate P'(9):

P'(x) = -6(9)² + 420(9)^-1/2

A negative fractional exponent might look strange here, so remember that a negative exponent sends that term to the denominator:

P'(x) = -6(9)² + 420/√9

P'(x) = -6*81 + 420/3

P'(x) = -6*81 + 420/3

P'(x) = -486 + 140

P'(x) = -346

In this problem, that means you're losing $346/hundred cases when you're producing 900 cases. Notice that this derivative is negative, which means the rate is decreasing.