Question

Here is a linear demand function: Q = 10 -0.5P. Find its price function by inverting the demand function. Then find its total revenue function by multiplying through by Q. The linear demand function Q = 400 -250P inverts into the price function P = 1.6 -0.004Q. Multiplying this by Q gives its total revenue function TR = 1.6Q -0.004. Evaluate the following expression.

Y = 5(2X + 3)2 -2X2

Answer 1

`P = 20 - 2Q`

Step-by-step explanation:

`Q = 10 - 0.5P`

Price function can be estimated by inverting the demand function.

`Q = 10 - 0.5P \\\\0.5P = 10 - Q\\P = 10/0.5 - Q/0.5 \\P = 20 - 2Q`

This is the price function.

Total revenue function can be estimated using the given formula,

`TR = P*Q \\ = (20 - 2Q) Q \\ = 20Q - 2Q^2`

The linear demand function is given by,

`Q = 400 - 250P \\`

Price function is given by,

`P = 1.6 - 0.004Q \\`

Total revenue function is thus given by,

`TR = P*Q \\ = 1.6Q - 0.004Q^2`

`Y = 5(2X+3)^2 - 2X^2 \\Y = 5(4X^2 + 9 + 12X) - 2X^2\\Y = 20X^2 + 45 + 60X - 2X^2\\Y = 18X^2 + 45 + 60X \\`

The derivative of Y with respect to x is,

`dY/dX = 36X + 60\\`

Equating this equal to 0 we get,

`36X + 60 = 0 \\36X = -60 \\X = -10/6 \\\\X= -1.66`