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)

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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.

(The entire section contains 190 words.)

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