The total cost, in dollars, to produce q items is given by the function C ( q ) = 41 , 000 + 12.50 q − 0.001 q 2 . Find the total cost of producing 500 items.

Question

Find the marginal cost when producing 500 items. That is, find the cost of producing the 501^st item.

Answer 1

Hi Jalal! There are two ways to solve this question, one using Calculus and one using basic Algebra.

As the question says, the marginal cost when producing 500 items means the same thing as the cost of producing the 501^st item.

One way to do it would be to take the derivative of the original cost function. This is because the derivative of the cost function is equal to the marginal cost function.

The cost function is C(q) = 41,000 + 12.50 q - 0.001q²

Here is the formula for taking the derivative, f'(x), of a term, f(x):

f(x) = ax^b

f'(x) = (a*b)x^b-1

Let's go through the process for each term:

For the first term:

41,000 is a constant so the derivative is 0. This is because the derivative of a constant is always 0.

Now for the second term:

f(q) = 12.50q¹ (If there is no exponent written, the exponent is assumed to be 1)

f'(q) = (12.50*1)q^1-1

f'(q) = 12.50q⁰

f'(q) = 12.50*1 (Anything to the power of zero is equal to 1)

f'(q) = 12.50

Now for the last term

g(q) = -0.001q²

g'(q) = (-0.001 * 2)q^2-1

g'(q) = (-0.002)q¹

g'(q) = -0.002q (When there is an exponent of 1, it does not need to be written)

Now we combine the derivatives of each term to get the full derivative of the original equation:

C'(q) = 0 + 12.50 - 0.002q

Thus, the derivative would be C'(q) = 12.50 - 0.002q

To find the marginal cost of producing 500 items we plug in 500 for q, which looks like this:

C'(q=500) = 12.50 - 0.002(500)

Then we simplify the expressions in the equation:

C'(q=500) = 12.50 - 1

Then we perform the subtraction

C'(q=500) = 11.50

Now we have the marginal cost of producing 500 items is $11.50

Another way to do this would be to find the cost it takes to make 501 items and the cost to make 500 items, then to subtract the second term from the first one. It may sound confusing written like this, so let's go through it together.

The cost function is C(q) = 41,000 + 12.50 q - 0.001q²

To find the cost of producing 501 items we plug in 501 for q, which looks like this:

C(q=501) = 41,000 + 12.50 (501) - 0.001(501)²

Then we simplify the expressions in the equation:

C(q=501) = 41,000 + 6,262.5 - 251.001

Then combine them all up:

C(q=501) = 47,011.499

Now we have the cost of producing 501 items is $47,011.499

Repeat the same process to find the cost for 500 items

The cost function is C(q) = 41,000 + 12.50 q - 0.001q²

To find the cost of producing 500 items we plug in 500 for q, which looks like this:

C(q=500) = 41,000 + 12.50 (500) - 0.001(500)²

Then we simplify the expressions in the equation:

C(q=500) = 41,000 + 6,250 - 250

Then add them all up:

C(q=500) = 47,000

Now we have the cost of producing 500 items is $47,000

Now to find the marginal cost we just find the difference in costs:

C(q=501) - C(q=500) = Marginal Cost

47,011.499 - 47,000 = Marginal Cost

11.499 = Marginal Cost

Rounding to the nearest cent the marginal cost when producing 500 units would be $11.50

As you can see, both methods reach the same answer.

Hope this helped!

-Allen