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The demand function for our product is: Q=90-0.1P^2

a.) Fine the elasticity when Q=50.

b.) Is the demand elastic or inelastic?

If we raise the price will our total revenue increase or will it decrease? Find the maximum revenue and the values of P and Q which will maximize the revenue.

Verified Answer :

Since3122015

Assuming that "elasticity" = P:

A. \(Q=90-0.1P^2\\50=90-0.1P^2\\-40=-0.1P^2\\400=P^2\\20=P\\\\\frac{dQ}{dP}=-0.2P\\E=\frac{P}{Q}*\frac{dQ}{dP}\\E=\frac{(20)}{(90)}(-0.2P)\\E=\frac{-2(20)}{45}\\E=\frac{-8}{9}\) - Elasticity = -0.889

B. \(0>\frac{-8}{9}>-1>-\infty\) - The demand is inelastic because the elasticity > -1.

(C). Set P and Q to 1 in two separate functions. If Q < P revenue will increase. If Q > P revenue will decrease.

\(Q=90-0.1P^2\\Q=90-0.1(1)^2\\Q=90-0.1\\Q=89.9\\\\(1)=90-0.1P^2\\-89=-0.1P^2\\890=P^2\\\sqrt{890}=P\\P=29.833\\Q>P\\(89.9)>(29.833)\)

Q > P therefore revenue will decrease.

(D). \(Q=90-0.1P^2\\\frac{dQ}{dP}=-0.2P\\-0.2P=0\\P=0\)

One obviously won't be able to maximize revenue if their price per unit, P, equals 0. Quantity of a product can only be sold in whole, so the closest integer to 90 is 89. The value of P that maximizes revenue is \(Q=90-0.1P^2\\(89)=90-0.1P^2\\-1=-0.1P^2\\10=P^2\\\sqrt{10}=P\) - sqrt(10).

Therefore, the values of P and Q that maximize revenue are 3.162 and 89, respectively.

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