Question

A company manufactures fuel tanks for cars. The total weekly cost (in dollar) of producing a tanks is given by

C(x)= 10000+90x-0.05x2
(a) 1. Find the marginal cost function.
(b) 2. Find the marginal cost at a producing level of 500 tanks/week..
(c) 3. Find the exact cost of producing the 501st item.
I

Answer 1

The marginal cost function is C'(x) = 90 - 0.1x. The marginal cost at a producing level of 500 tanks/week is $40. The exact cost of producing the 501st item is $47049.5.

Step-by-step explanation:

To find the marginal cost function, we need to find the derivative of the total cost function with respect to the quantity produced, x.

So, taking the derivative of C(x) = 10000 + 90x - 0.05x^2:

C'(x) = 90 - 0.1x

Hence, the marginal cost function is C'(x) = 90 - 0.1x.

To find the marginal cost at a producing level of 500 tanks/week, we substitute x = 500 into the marginal cost function:

C'(500) = 90 - 0.1(500) = 90 -50 = 40.

Therefore, the marginal cost at a producing level of 500 tanks/week is $40.

To find the exact cost of producing the 501st item, we substitute x = 501 into the total cost function:

C(501) = 10000 + 90(501) - 0.05(501)^2 = 47049.5.

Therefore, the exact cost of producing the 501st item is $47049.5.

Answer 2

To find the marginal cost function, differentiate the total cost function. To find the marginal cost at a specific level, substitute the value into the marginal cost function. To find the exact cost of producing a specific item, substitute the value into the total cost function.

Step-by-step explanation:

To find the marginal cost function, we need to differentiate the total cost function with respect to the producing level x. Let's differentiate the given cost function, C(x) = 10000 + 90x - 0.05x², using the power rule. The derivative will give us the marginal cost function. After finding the marginal cost function, we can substitute x = 500 to find the marginal cost at a producing level of 500 tanks/week. To find the exact cost of producing the 501st item, we can substitute x = 501 into the total cost function C(x).