Math help wanted :)

[marie7]

A firm’s cost function and the demand functions are
C(x) = 5x and p = 25-2x respectively, where x is the amount produced and/or demanded.

(a) Find the output level that will maximize the firm’s profits. What is the
maximum profit?

(b) If a tax of t per unit is imposed, which the firm adds to its cost, find the output level that will maximize the firm’s profits. What is the maximum profit? (Note that the optimal output and profits will both be functions of the tax rate t rather than some fixed values.)

(c) Determine the tax t per unit that must be imposed to obtain the maximum tax revenue. (Hint: Use the solution of output as a function of t from 2(b) to formulate the tax revenue function.)

(d) Given the solution for t in (c), find the optimal output and profits using their optimal choice functions in (b). (This time, the solutions will be some fixed values.)

I have absolutely no clue how to do these - could someone show me how to do these please?

 

[mmm4444bot]

In part (A), the Demand function (usually written as D(x) = p) gives the price at which x units will sell.

We're given an expression for p in terms of the number of units sold: 25 - 2x.

Revenue is obtained by multiplying the number of units sold by the price.

Therefore, the Revenue function is R(x) = x * p. This simplifies.

R(x) = 25x - 2x^2

Profit is Revenue minus Costs.

P(x) = R(x) - C(x)

This simplifies.

P(x) = -2x^2 + 20x

Are you familiar with quadratic models, parabolas, vertex, related formulas, et cetera?

If you need more help for part (a), then please ask specific questions.

 

[marie7]

Yes I understand parabolas and the vertex stuff - so what's the answer to a) overall?
I would also like some help with the other parts of the question please
Thank you so much for your help

 

[mmm4444bot]

… I understand … so what's the answer …

Good grief!

Is your expectation here that people will do your homework for you? If so, then you've come to the wrong place.

You show us your reasoning on part (a), and then we'll be able to determine whether or not your self-claimed understanding is sufficient.

 

[marie7]

Re:

… I understand … so what's the answer …

Good grief!

Is your expectation here that people will do your homework for you? If so, then you've come to the wrong place.

You show us your reasoning on part (a), and then we'll be able to determine whether or not your self-claimed understanding is sufficient.

Yes but what am I expected to do from "P(x) = -2x^2 + 20x"? just tell me, don't say the answer, and I'll figure it out.

 

[marie7]

1st derivative -4x +20
when -4x + 20 = 0 is when x = 5
Is this what I'm expected to do, find derivatives?

 

[mmm4444bot]

Part (a) asks for the value of x that maximizes the value of P(x).

The maximum value of P(x) occurs at the vertex point of the parabola that is the graph of function P.

Using derivatives is one way to find the value of x at the vertex. I have no idea whether you're "expected" to use any particular approach.

The formula for the x-coordinate of the vertex of a parabola is -b/(2a), so that would work also.

x = 5 is the correct answer for the first question in part (a).

Do you understand the second question, in part (a)?