f.o.i.l word problem

[needhelpplease1]

A company owns 190 storage units which are fully used when the rent is 100 per month. For every 10 increase in rent 5 units will become vacant. What rent should be charged in order to obtain the largest gross income?

so far I have
(100 + 10x) (190 - 5x)
19000 + 1400x -500x^2

Havnt done this in a while so not sure if my answer is correct or if im approacing this the right way. Thanks for any help.

 

[lookagain]

A company owns 190 storage units which are fully used when the rent is 100 per month. For every 10 increase in rent 5 units will become vacant. What rent should be charged in order to obtain the largest gross income?

so far I have
(100 + 10x) (190 - 5x)
19000 + 1400x -500x^2

Havnt done this in a while so not sure if my answer is correct or if im approacing this the right way. Thanks for any help.

I am used to doing this type of problem in a calculus class.

In algebra, notice that this is a quadratic expression.

For \(\displaystyle ax^2 + bx + c = 0,\) the extreme point occurs at \(\displaystyle x = \frac{-b}{2a}.\)

Rearrange what you have in this form and set it equal to zero:

\(\displaystyle -500x^2 + 1400x + 19000 = 0\)

\(\displaystyle a = -500\)

\(\displaystyle b = 1400\)

\(\displaystyle x = \frac{-(1400)}{2(-500)}\)


\(\displaystyle x = \frac{7}{5} \ or \ 1.4\)

In the rent expression that you worked out, you have that the rent corresponds to \(\displaystyle 100 + 10x.\)

Plug in \(\displaystyle x = 1.4\) into this rent expression.

\(\displaystyle 100 + 10(1.4) = 100 + 14 = 114\)


This corresponds to a rent of \(\displaystyle \boxed{\$ 114}\) in order to obtain the largest gross income.

 

[needhelpplease1]

Thank you lookagain.