Calculus Word Problem helper?

Question

One more word problem, please help me:

The profit (in dollars) from the expenditure of x thousand dollars on advertising is given by P(x) = 1000 + 32x - 2x^2. Find Marginal profit at following expenditures. Then in each case, say whether firm should increase the expenditure and why.

a.) x = 8

b) x = 6

c) x = 20


Please don't tell me its easy, it is hard for me. Please show work and explain how do it.

Answer 1

Question a)
P(8) = 1000 + (32 x 8) - (2 x 64)
P(8) = 1000 + 256 - 128
P(8) = 1128
marginal profit = $128

Question b)
P(6) = 1000+ (32 x 6) - (2 x 36)
P(6) = 1000 + 192 - 72
P(6) = 1120
marginal profit = $120

Question c)
P(20) = 1000 + (32 x 40) - (2 x 400)
P(20) = 1000 + 1280 - 800
P(20) = 1480
marginal profit = $480

P(x) = 1000 + 32x - 2x²
P `(x) = 32 - 4x = 0
turning point
x = 8
Break even point for advertising is $ 8000

Answer 2

The question is asking for "marginal profit" - whenever you see "marginal", think slope, and then think derivitive (which gives you the formula for the slope).

First step is to take the derivitive of the profit formula - take each term separately. The "1000" goes away, the derivitive of "32x" is "32" and the dervitive of "2x^2" is "4x".

So, P'(x) = 32 - 4x ... this is your "marginal profit", and if you plug each of the options into this formula, you'll know if the firm should increase expenditures or not.

P'(8) = 0, which means you're at a maximum or minimum. In this case, it's a maximum (the overall graph is a downward opening parabola, so it will have a maximum but no minimum). The firm should not invest any more money.

P'(6) = 8, which is positive, which means that each additional dollar expended results in more profit. The firm should spend more.

P'(20) = -48, which is negative, which means each additional dollar results in LESS profit. The firm should definitely not spend more.

Looking at all of the options, the firm should come to the conclusion that spening $8000 is the wises course of action to maximize profit.

You can check this by plugging '8' into the original profit formula and then seeing that any other number you plug in will give you a lower profit.

Answer 3

substitute the values given for x in P(x) to find marginal profit

(a) x = 8
1000 + 32(8) -2(8)(8) =
1000 + 256 - 128 =
$1128

(b) x = 6
1000 + 32(6) - 2(6)(6) =
1000 + 192 - 72 =
$1120

(c) x = 20
1000 + 32(20) - 2(20)(20) =
1000 + 640 - 800 =
$840

To find the optimal expenditure, take the derivative of P(x) and set it equal to zero:

P(x) = 1000 + 32x - 2x^2
P'(x) = 32 - 4x
32 - 4x = 0
4x = 32
x = 8

So x=8 is the optimal expenditure. Any more or less than this is not cost effective.

Best of luck!

Answer 4

Profit dollar change with respect to advertising dollars invested.
When x is 8, Profit is 128
When x is 6, profit is 120
When x is 20, there is a loss -160

Draw the line and note that max profit occurs at x=8 so spend more on product b and stop spending on c, leave a alone.

Answer 5

Word Problem Helper