Question

6. A lighting fixture manufacturer has daily production costs of c=0.25n²-10n+800, where C is the total
daily cost in dollars and n is the number of light fixtures produced.
a) Is the manufacturer's cost increasing or decreasing when they produce between 10 and 15 light fixtures?
Prove your claim with math. (2 pts)
b) Is the manufacturer's cost increasing or decreasing when they produce between 20 and 25 light fixtures?
Prove your claim with math. (2 pts)

Answer 1

By finding the average rate of change, we can see that:

a) The cost decreases.

b) The cost increases.

How to know when the cost is increasing or decreasing?

To check that, we need to find the average rate of change on the interval.

Remember that for function f(x) on an interval (a, b), the average rate of change is:

R = (f(b) - f(a))/(b - a)

Here the cost function is:

c(n) = 0.25*n² - 10n + 800

a) In the interval [10, 15] the average rate of change is given by:

R = (c(15) - c(10)/(15 - 10)

Where:

c(15) = 0.25*15^2 - 10*15 + 800 = 706.25

c(10) = 0.25*10^2 - 10*10 + 800 = 725

Then the average rate of change is:

R = (706.25 - 725)/(15 - 10) = -3.75

This means that between 10 and 15 light fixtures, the cost is decreasing.

b) Now we have the interval [20, 25], so let's do the same ting:

c(20) = 0.25*20^2 - 10*20 + 800 = 700

c(25) = 0.25*25^2  - 10*25 + 800 = 706.25

Here the average rate of change is:

R = (706.25 - 700)/(25 - 20) = 1.25

It is positive, which means that the cost is increasing.

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