Math

QuestionThe demand function is D(x)=96725xD(x)=967-25-x.
(a) Find the elasticity of demand. (b) Find the price where elasticity equals 1. (c) Determine the price for maximum revenue.

Studdy Solution

STEP 1

Assumptions1. The demand function is given by (x)=96725x(x)=967-25-x, where xx is the price of the product in dollars, and (x)(x) is the demand in thousands of pieces. . The elasticity of demand is given by (x)=ddxx(x)(x) = \frac{d}{dx} \cdot \frac{x}{(x)}, where d/dxd/dx is the derivative of the demand function with respect to xx.
3. The price for which the elasticity equals1 is found by setting (x)=1(x) =1 and solving for xx.
4. The price yielding the maximum revenue is found by maximizing the revenue function R(x)=xD(x)R(x) = x \cdot D(x).

STEP 2

First, we need to find the derivative of the demand function with respect to xx. This will give us d/dxd/dx.
ddx=ddx(96725x)\frac{d}{dx} = \frac{d}{dx}(967-25-x)

STEP 3

Calculate the derivative.
ddx=1\frac{d}{dx} = -1

STEP 4

Now, we can calculate the elasticity of demand (x)(x) using the formula.
(x)=ddxx(x)(x) = \frac{d}{dx} \cdot \frac{x}{(x)}

STEP 5

Plug in the values for d/dxd/dx, xx, and (x)(x) into the formula.
(x)=1x96725x(x) = -1 \cdot \frac{x}{967-25-x}

STEP 6

This is the elasticity of demand function. Now, to find the price for which the elasticity equals1, we set (x)=1(x) =1 and solve for xx.
1x96725x=1-1 \cdot \frac{x}{967-25-x} =1

STEP 7

olving the above equation for xx gives us the price for which the elasticity equals1.
x=96725xx =967-25-x

STEP 8

olving the above equation gives us x=471x =471.

STEP 9

Now, to find the price yielding the maximum revenue, we need to maximize the revenue function R(x)=xD(x)R(x) = x \cdot D(x).

STEP 10

First, we plug in the values for xx and (x)(x) into the revenue function.
R(x)=x(96725x)R(x) = x \cdot (967-25-x)

STEP 11

To find the maximum of this function, we take the derivative of the revenue function with respect to xx and set it equal to zero.
dRdx=ddx(x(96725x))=0\frac{dR}{dx} = \frac{d}{dx}(x \cdot (967-25-x)) =0

STEP 12

olving the above equation gives us the price yielding the maximum revenue.
The solution is(a) The elasticity of demand is (x)=x96725x(x) = - \cdot \frac{x}{967-25-x}. (b) The price for which the elasticity equals is x=471x =471. (c) The price yielding the maximum revenue is found by solving the equation dRdx=0\frac{dR}{dx} =0.

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