# Calculus Word Problems

• May 14th 2007, 04:38 AM
janvdl
Calculus Word Problems
A hotel has 72 rooms.
Rooms are usually \$90 per night.
But if the owner raises the price to \$95 per room per night, he finds that there will be 2 rooms fewer than usual(\$90 a room per night) hired.

I am not sure whether for EVERY \$95 dollar room, 2 rooms are left vacant, or are 2 rooms of the 72 rooms in total left vacant.

I think our teacher said that the Income formula that we have to calculate is:
Income = 108x - [or maybe +] (2x^2)/5
-----------------------------

1. Find the formula to calculate the income

2. Calculate the maximum Income

------------------------------

Could someone please explain calculus word problems like this to me in detail?

Thank You! :)
• May 14th 2007, 05:26 AM
earboth
Quote:

Originally Posted by janvdl
A hotel has 72 rooms.
Rooms are usually \$90 per night.
But if the owner raises the price to \$95 per room per night, he finds that there will be 2 rooms fewer than usual(\$90 a room per night) hired.
...

1. Find the formula to calculate the income

2. Calculate the maximum Income

Hello,

I assume:
a) that each room is rented for \$90
b) that the price could be increased by multiple of \$5:

let x be the times that the price is increased:
let i be the income:

i(x) = (72 - 2x)(90 - 5x) = -10x² + 180x + 6480

The original situation is i(0) = \$6480

The graph of this function is a parabola opening downward so the maximum value is at it's vertex. Use derivation:
i'(x) = -20x + 180. You'll get the maximum if i'(x) = 0 ==> x = 9

i(9) = \$7290 when only 72 - 18 = 54 rooms are rented for a price of \$(90-5*9) = \$45
• May 14th 2007, 05:31 AM
janvdl
Sorry Earboth, my mistake. You are correct in what you did, but it still doesnt give me the income formula that I should have got...
• May 14th 2007, 05:41 AM
Soroban
Hello, janvdl!

The problem is badly worded, but I'm familiar with this type.

Quote:

A hotel has 72 rooms. .Rooms are usually \$90 per night.
But if the owner raises the price to \$95 per room per night,
he finds that there will be 2 fewer rooms rented than usual.

(1) Find the formula to calculate the income
(2) Calculate the maximum Income

The problem is usually stated like this . . .

When the owner charges \$90 per room, all 72 rooms are rented.
Then he finds that for every \$5 increase in the rate, he loses 2 guests.

Let x = number of five-dollar increases in the rate.

The rate will be: .90 + 5x dollars per night.

He will lose 2 guests for every increase.
. . Hence, he will have only: .72 - 2x guests.

His income will be: .(90 + 5x)(72 - 2x) dollars.

Therefore, the income function is: .I .= .6480 + 180x - 10x² .(1)

Maximize the income function: .I' = 0

We have: .180 - 20x .= .0 . . . . x = 9 .**

The maximum income is: .I .= .6480 + 180·9 - 10·9² .= .\$7290 .(2)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

**

To maximize his income, the owner should raise the rate nine times.
He will charge: .90 + 9·5 .= .\$135 per night.

As a result, he will lose 9·2 = 18 guests.
. . He will have only: 72 - 18 .= .54 guests.

But he will receive: .\$135 × 54 .= .\$7290

[Originally, his income was: .\$90 × 72 .= .\$6450]

• May 14th 2007, 05:48 AM
janvdl
Ah, ok i understand now.

Thanks a lot to both of you! :)