Maximum and Minimum Question

1. May 30, 2010

Procrastinate

What are the dimensions of the largest rectangle that can be drawn in the closed region bounded by the x- and y-axes and the graph of the function y = 8 - x3
A=lw=16
D= 2l + 2w = 2l + 16/l

I started off by graphing the equation. The y-int was 8 and the x-int was 2.
Thus, I established that the total area bound by the axes was 16.
Afterwards, I found the stationary points of D:

D'=2-32/l2
D''=-64/l2
The stationary point was four and thus as D'' was a negative, D is a maximum. L=4 and thus w=4 and the total dimensions were 16.

However, this was wrong, the answer was 15. I think I may have interpreted the question incorrectly but could someone please point out my error? Thank you.

Last edited: May 30, 2010
2. May 30, 2010

Mentallic

I think I may have interpreted the question incorrectly also... I first assumed "the largest rectangle" to be the one with most area. From what you've done I think you're assuming it's the largest perimeter.

The total area bounded by the axes and the function is not 2x8=16, it's $$\int_0^2ydx$$ where $y=8-x^3$

For the largest area, I found an answer of 6.21/3 which have dimensions x=21/3 and y=6. For the largest perimeter, I have x=1/31/2.

So I'm at a complete loss of how the book gets the answer of 15...

3. May 30, 2010

cronxeh

P = 2x + 2y
A = xy
Maximize P.

I think you can only get 15 as the answer if you find maximum perimeter here. Something like x~1 and y~6.5., P~2+13=15

Although x=1 and y=7 give you greater perimeter? I don't get how the book gets 15! Maybe he rotated the rectangle like 45 degrees..

Last edited: May 30, 2010
4. May 30, 2010

ƒ(x)

shouldnt you maximize area?

5. May 31, 2010

Mentallic

Procrastinate, can you please be more clear on what the answer from the book is. "15" on its own doesn't tell us much, and makes things especially confusing when the question itself can be interpreted in a thousand different ways.

6. Jun 1, 2010

Procrastinate

The answer just says 15 and doesn't provide any further explanation, unfortunately.

7. Jun 1, 2010

Mentallic

Ok well can you at least tell us what your mathematical background is? If the answer is looking for a rotated rectangle as cronxeh suggested, then this is obviously going to be a lot more complicated than the average calculus 1 student can do.

8. Jun 1, 2010

Mentallic

And can I just point out something,

w=16/l, therefore D=2l+2(16/l)=2l+32/l

not what you have.

9. Jun 2, 2010

Procrastinate

High school maths student.

10. Jun 2, 2010

Mentallic

Then it's safe to say that we are restrained to the rectangle being "parallel" to the axes. i.e. not turned at some angle.

Well, this question still bothers me...

wow...