The Cool Company determines that the supply function for its basic air conditioning unit is S (p) = 40 + 0.008p^3 and that its demand function is D (p) = 200 - 0.16p^2, where p is the price. Determine the price for which the supply equals the demand.

In this item, we are to determine the price to which there would be a breakeven between the supply and demand. This can be done by equating the equations given for each as shown below.

S (p) = D (p)

40 + 0.008p³ = 200 - 0.16p²

We transpose all the terms to left side of the equation.

0.008p³ + 0.16p² - 160 = 0

Then, we factor out the given function. This can either be done through long division or synthetic division. Divide the equation by 0.008.

p³ + 20p² - 20000 = 0

One of the roots of the equation is 21.86. This is the only real root of the equation and the rest are imaginary.

Hence, the price should be approximately $21.86 before the demand and supply is breakeven.