Word Problem involving Optimization

[919_is]

PROBLEM: A 300-room hotel in Las Vegas is filled to capacity every night at $80.00 a room. For each $1.00 increase in rent, 3 fewer rooms are rented. If each rented room costs $10.00 to service per day, how much should the management charge for each room to maximize gross profit? What is the maximum gross profit?

ANSWER:
Here is what I have so far:

Let x = number of dollar increases in the rent per night

300-3x = total number of rooms rented
80+x = rent per night


y(x) = (300-3x) - 10 (80+x)
y(x) = -13x-500

y'(x) = -13

Therefore, no critical value.

Is this answer correct? No critical value means no local maximum or minimum, right?

Please advise. I am not exactly sure how else to proceed.

Thanks in advance.

 

[tkhunny]

Try that again.

y(x) = (300-3x)*(80+x) - (300-3x)*10

You were just counting the rooms, not actually charging for them.

 

[919_is]

Aahh ... thank you so much for that!

Here is a second attempt:

Let x = number of dollar increases in the rent per night

300-3x = total number of rooms rented
80+x = rent per night


y(x) = (300-3x)*(80+x) - (10)*(80+x)
y(x) = -3(x)^2+90x+21000

y'(x) = -6x+90

x= $15.00 increase per night

Critical Value: x = 15

y(15) = [(300-(3*15))*(80+15)] - [(10)*(80+15)]
y(15) = [255*95] - [10*95]
y(15) = 24225 - 950
y(15) = 23275

What is the maximum gross profit? $23,275

y"(x) = -6 < 0

Absolute Maximum: x = 15

80+x = ?
80+15 = 95

How much should the management charge for each room to maximize gross profit? $95 per night

Is this correct?

 

[mmm4444bot]

It's not yet correct.

What do you think the expression (10)*(80+x) represents?

Good job, defining your symbol x to start, btw. :cool:

 

[919_is]

It's not yet correct.

What do you think the expression (10)*(80+x) represents?

Good job, defining your symbol x to start, btw. :cool:

That should be my cost ... and I think this is where I made the mistake .... it should have been (10)*(300-3x) Thank you for pointing that out. It does get confusing and tedious.

Ok, here it goes ....

Let x = number of dollar increases in the rent per night

300-3x = total number of rooms rented
80+x = rent per night


y(x) = (300-3x)*(80+x) - (10)*(300-3x)
y(x) = -3(x)^2+90x+21000

y'(x) = -6x+90

x= $15.00 increase per night

Critical Value: x = 15

y(15) = [(300-(3*15))*(80+15)] - [(10)*(300-(3*15))]
y(15) = [255*95] - [10*255]
y(15) = 24225 - 2550
y(15) = 21675

What is the maximum gross profit? $21,675

y"(x) = -6 < 0

Absolute Maximum: x = 15

80+x = ?
80+15 = 95

How much should the management charge for each room to maximize gross profit? $95 per night

I hope this works.

 

[tkhunny]

y(x) = (300-3x)*(80+x) - (10)*(80+x)
y(x) = -3(x)^2+90x+21000

(300-3x)*(80+x) - 10*(300-3x) =

24000 + 300x - 240x - 3x^2 - 3000 + 30x =

21000 + 90x - 3x^2

 

[919_is]

It kind of makes me angry when the algebra gets in the way of the calculus. Generally, I blame high school for their panic attitude to get folks to the calculus. Probably everyone could use a little more time in algebra.

(300-3x)*(80+x) - 10*(80+x) =

24000 + 300x - 240x - 3x^2 - 800 - 10x =

23200 + 50x - 3x^2.

Actually, that is my fault. I did my computation on paper, and when I posted it, I mistyped the original formula with the correct computation. It should have been:


y(x) = (300-3x)*(80+x) - 10*(300-3x)
y(x) = 24000 + 300x - 240x - 3x^2 - 3000 - 30x
y(x) = 21000 + 90x - 3x^2


Sorry, about the confusion ....


y'(x) = -6x+90

x= $15.00 increase per night

y"(x) = -6 < 0

Absolute Maximum: x = 15

y(15) = [(300-(3*15))*(80+15)] - [(10)*(300-(3*15))]
y(15) = [255*95] - [10*255]
y(15) = 24225 - 2550
y(15) = 21675

 

[mmm4444bot]

What is the maximum gross profit? $21,675

How much should the management charge for each room to maximize gross profit? $95 per night

Looks good.

If you're familiar with the graphs of quadratic equations (y = a*x^2 + b*x + c), then you know that they are parabolas. Any maximum will occur at the vertex. In terms of the coefficients a, b, and c, the x-coordinate of the vertex is always -b/(2a). That's for a check. :cool:

 

[919_is]

Thank you for the help and the tips!