# A diamond dealer finds that the demand for flawless diamonds is governed by `p=4x^2-60x+20` , where x is in carats (in units of hundred) with `0ltxlt=5` and p is in thousands of dollars.  Find the...

A diamond dealer finds that the demand for flawless diamonds is governed by `p=4x^2-60x+20` , where x is in carats (in units of hundred) with `0ltxlt=5` and p is in thousands of dollars.

Find the number of carats to be made available to maximize revenue.

The number of carats is _____ (in units of hundred).

What are the price and the revenue for this amount?

The price is ______ (in thousands of dollars)
and the revenue is ______ (in thousands of dollars).
(For the price and the revenue, you may found the calculations easier using a calculator)

lemjay | Certified Educator

calendarEducator since 2012

starTop subjects are Math and Science

Note that revenue is the product of the demand function and the number of units sold. So, our revenue function is:

`R = p*x=(4x^2-60x+20)x`

`R = 4x^3-60x^2+20x`

To determine the number of carats (x) that will maximize the revenue, take the derivative of R.

`R' = 12x^2-120x+20`

Set R' equal to zero.

`0=12x^2-120x+20`

Divide the equation by the GCF of 12,20 and 120, to simplify.

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