Optimization help!

[zodiacbrave]

There is this problem from my textbook but I can't seem to figure it out.

 

A rectangular swimming pool is to be built with the area of the pool being 2500 square feet. In addition, the owner wants 6 foot wide decks along the sides of the pool and 8 foot wide decks at the two ends. Determine the dimensions (to the nearest tenth of a foot) of the pool that allows the pool and decks to be built on the smallest possible piece of property.

 

Thank you

[timo]

Reduce the two variables to one leftover variable by substituting one of the two equations you get into the other. Then find the remaining variable such that it minimizes the leftover function for the total area. Not completely sure if that works (I did not actually try the problem) but it should.

 

If you have any intermediate results yourself and just got stuck, please show/post what you already tried and where and why you got stuck.

[zodiacbrave]

I'm having a hard time really visualizing this problem. From what I understand, the pool's area itself is 2500. Along with the area of the pool, there are decks that wrap around the perimeter of the pool. At least I believe it wraps around the entire way around.

 

 

 

So if the pool area itself is 2500, and I want to find the dimensions of the smallest amount of area that both the pool and the deck occupies. if x is the width and y is the length of the pool and xy= 2500, then A=2500.

 

The real problem that I'm having is the fact that the pool deck's width's are given but not the lengths. I'd assume the length's would also be the length and width of the pool.

 

2500 + width of one rectangle*lengthofpool + width of other rectangle*width_of_pool = total area of both the pool and the decking?

 

The book asks to solve by graphing. So I'd assume i would need a parabola

[timo]

Sadly, I don't know what a "deck" is. What you assumed is that it's two rectangles being on one of the longer and one of the shorter sides, respectively. I could imagine it's either four of them (one on each side of the pool) in which case you are missing two addends for the total area (two additional "decks") or kind of a ring around the pool in which case you must add four additional "corner-rectangles" to fully enclose the pool. Except for that possible problem, your approach seem fine so far.

 

Regardless of the actual term you use to calculate the total area (i.e. regardless of what additional area you decide to take for the "decks") from what you have you can solve the problem in the manner I explained earlier:

Substitute either the lengths of the pool or the width of the pool with the respective other variable (note that xy=2500) to get expression for the total area that only depends on one of the two variables. Then, find the minimum total area as a function of this variable, either by differentiation or by drawing the function.

[D H]

Athiest: a picture of a pool with a deck:

 

Zodiac: I picture is worth a thousand words, so draw a picture of the pool with the decks around it. Make the pool rectangular. Call the longer edges of the pool the "sides" and the shorter edges the "ends". (Or vice versa, but I think my nomenclature is in line with the vernacular.)

 

Regarding the decks: You are hung up on the words "length" and "width". Each deck extends a bit from the edge of the pool. Suppose the pool has dimensions 100 feet by 25 feet (THIS IS NOT THE ANSWER, SO DON'T USE IT). Suppose you are standing on a deck alongside the long dimension. How wide is the deck? (Answer: 6 feet. It is 100 feet long). Now imagine walking to a deck along the end of the pool. That deck is 8 feet wide, not 25 feet.

 

BTW, whether those four 6'x8' rectangular chunks at the corners of the decks count or don't count is irrelevant. They're total area is 192 sq. ft., a constant.

[Mr Skeptic]

So then the area of the pool&decks is A = (6 + length + 6)(8 + width + 8) = (length + 12)(width +16), where width*length = 2500. So you replace either length or width with a function of the other, and optimize by whatever means you can. One way to optimize is to find where the derivative is zero, or you could do a graph.