Question

A company manufactures mountain bikes. The research department produced the marginal cost function Upper C prime​(x)equals700- x/ 3

​, 0 less than or equals x less than or equals 900​, where Upper C prime​(x) is in dollars and x is the number of bikes produced per month. Compute the increase in cost going from a production level of 450 bikes per month to 900 bikes per month. Set up a definite integral and evaluate it.
The increase in cost is ​$
nothing.

Answer 1

The increase in cost is
`C _((900 - 450 )) =` $ 213750

Explanation:

From the question we are told that

The marginal cost function is
`C \ '(x) = 700 -(x)/(3)` where
`0 \le x \le 900`

Since
`C \ '(x)` is in dollars it means that
`C(x)` is the cost price and x is given as the number of bikes now to obtain
`C(x)` for the given increase of the number of bikes we integrate
`C\ '(x)` within the increase limit

This can be evaluated as

`\int\limits^(900)_(450) {C \ '(x)} \, dx = \int\limits^(900)_(450) {700 - (x)/(3) } \, dx`

`C(x) = [ {700x - (x^2)/(6) }] \left 900} \atop {450}} \right.`

`C _((900 - 450 )) = [700(900) - (900^2)/(6) ] - [700(450) - (450^2)/(6) ]`

`C _((900 - 450 )) = 495000- 281250`

`C _((900 - 450 )) =` $ 213750