Hello, isabelle2hot!
Since you posted this under "Beginning Alegebra",
. . I assume we're not to use Calculus.
. . I'll baby-talk through it for you . . .
You manage a 160 unit motel.
All units will be occupied, on average, when you charge $50 per day per unit.
Experience shows that for each $1 per day per unit that you increase the price, you will accrue 2 vacancies.
Each occupied room costs $4 per day to clean and supply.
On this basis, what price, in dollars, should you charge per unit per day to maximize your profit?
a. $87 . b. $82 . c. $77 . d. $72 . e. $67 . f. $62 . g. $57 . h. none of these
Right now, we charge $50/day and have all 160 units occupied.
. . Our revenue (income) is: \(\displaystyle 160\,\times\,\$50\:=\:\$8,000\) per day.
. . Each room costs $4 to clean: cost = \(\displaystyle \$4\,\times\,160\:=\:\$640\)
Our profit is:
.\(\displaystyle P\:=\:\$8,000\,-\,640\:=\:\$7,360\) per day . . . Can we do any better?
Let \(\displaystyle x\) = number of $1 increases in the rate.
. . The new rate is: \(\displaystyle r\,=\,50+x\) dollars per unit.
For each $1 increase, we lose 2 customers.
. . The number of occupied units is: \(\displaystyle N\,=\,160-2x\)
Our revenue will be: \(\displaystyle R\;=\;r\cdot N\:=\
50+x)(160-2x)\) dollars.
But each of the \(\displaystyle N\) rooms cost $4 to clean.
. . \(\displaystyle C\:=\:4(160-2x)\) dollars.
Since "Profit = Revenue - Cost", we have our function!
. . \(\displaystyle P\:=\
50\,+\,x)(160\,-\,2x)\,-\,4(160\,-\,2x)\)
. . which simplifies to:
.\(\displaystyle P\:=\:7360\,+\,68x\,-\,2x^2\)
But what \(\displaystyle x\) produces maximum \(\displaystyle P\)?
Note that the function is a quadratic.
. . The graph is a down-opening parabola.
. . The maximum occurs at its
vertex.
For the parabola: \(\displaystyle y\:=\:ax^2\,+\,bx\,+\,c\), the vertex is at: \(\displaystyle x\,=\,\frac{-b}{2a}\)
Our parabola has: \(\displaystyle a=-2,\,b=68,\,c=7360\)
. . Its vertex is at: \(\displaystyle x\,=\,\frac{-68}{2(-2)}\,=\,17\)
Therefore, we should have 17 increases in the rate.
. . The new rate is: \(\displaystyle \$50\,+\,17\:=\:\$67\) per day . . . answer (e)
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
More baby-talk . . .
We raise the rate by $17 ... to $67 per day
. . then \(\displaystyle 2\,\times\,17\,=\,34\) units will be empty.
. . We will have only: \(\displaystyle 160\,-\,34\,=\,126\) units occupied.
Our revenue is: \(\displaystyle \$67\,\times 126\:=\:\$8,442\) per day.
The 126 rooms cost: \(\displaystyle 126\,\times \$4\:=\:\$504\) to clean.
Our profit is: \(\displaystyle \$8,442\,-\,504\:=\:$7938\) per day.
. . Yes!
.This is better than our original $7,360 per day.
