The total cost, in dollars, to produce q items is given by the function C ( q ) = 54 , 000 + 14.50 q . Find the total cost of producing 800 items.

Question

part two Find the marginal cost when producing 800 items. That is, find the cost of producing the 801^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 800 items means the same thing as the cost of producing the 801^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) = 54,000 + 14.50 q

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:

54,000 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) = 14.50q¹ (If there is no exponent written, the exponent is assumed to be 1)

f'(q) = (14.50*1)q^1-1

f'(q) = 14.50q⁰

f'(q) = 14.50*1 (Anything to the power of zero is equal to 1)

f'(q) = 14.50

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

C'(q) = 0 + 14.50

Thus, the derivative would be C'(q) = 14.50

Since the marginal cost function is equal to a constant number with no variables. The marginal cost will always be that constant number no matter what quantity of items is produced.

In this case the marginal cost at all quantities will be $14.50

Now we have the marginal cost of producing 800 items is $14.50

Another way to do this would be to find the cost it takes to make 801 items and the cost to make 800 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) = 54,000 + 14.50 q

To find the cost of producing 801 items we plug in 801 for q, which looks like this:

C(q=801) = 54,000 + 14.50 (801)

Then we simplify the expressions in the equation:

C(q=801) = 54,000 + 11,614.5

Then combine the two terms:

C(q=801) = 65,614.5

Now we have the cost of producing 801 items is $65,614.50

Repeat the same process to find the cost for 800 items

The cost function is C(q) = 54,000 + 14.50 q

To find the cost of producing 800 items we plug in 800 for q, which looks like this:

C(q=800) = 54,000 + 14.50 (800)

Then we simplify the expressions in the equation:

C(q=800) = 54,000 + 11,600

Then add them all up:

C(q=500) = 65,600

Now we have the cost of producing 500 items is $65,600

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

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

65,614.50 - 65,600 = Marginal Cost

14.50 = Marginal Cost

The marginal cost when producing 800 units would be $14.50

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

Hope this helped!

-Allen