Help! A call to all those optimization whizzes.

Oct 2008
1
0
A rectangular swimming pool is to be built with an area of 1800 square feet. The owner wants 5-foot wide decks along either side and 10-foot wide decks at the two ends. Find the dimensions of the smallest piece of property on which the pool can be built satisfying these conditions.

I don't want to say all the ways i've tried to solve this thing for fear it'll get everyone stuck on the wrong train of thought. But be assured I've tried every way I can think of to get this problem in terms of x and solve for the minimum area of the pool, aaand i can't seem to get anything to work. A whole lot of gratitude and a lot of props to whoever can solve this.
 
Aug 2007
3,171
860
USA
Kali, Kali... How can we help if you show no work? Bad call.

Rule #1 - Name stuff

x = length of pool sides
y = length of pool ends

It's rectangular, so both ends and both sides arr the same, right?

Now Translate!

"A rectangular swimming pool is to be built with an area of 1800 square feet"

x*y = 1800 - Done!

"The owner wants 5-foot wide decks along either side "

5 + y + 5 = 10 + y = length of property ends

"10-foot wide decks at the two ends"

10 + x + 10 = x + 20 = length of property sides

Note: If you have not yet created a drawing to demonstrate this, please do so now. You must be convinced that the measurements are correct. If it should be y + 20 and x + 10, we're barking up the wrong tree.

"Find the dimensions of the smallest piece of property "

The area of the property is A(x,y) = (10+y)(x+20)

As the two variables are a little irritating, perhaps, we have a provision to get rid of one of them. Remember this x*y = 1800? Solve for y.

y = 1800/x

Substitute into the Area function, giving A(x) = (10 + 1800/x)(x+20)

Where does that leave us?
 

Soroban

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)
Hello, Kali!

A rectangular swimming pool is to be built with an area of 1800 ft².
The owner wants 5-ft wide decks along either side and 10-ft wide decks at the two ends.
Find the dimensions of the smallest piece of property on which this pool can be built.
Code:
      :  10 : - - - L - - - :  10 : 
    - * - - * - - - - - - - * - - *  -
    5 |                           |  :
    - *     * - - - - - - - *     *  :
    : |     |               |     |  :
    : |     |               |     |  :
    W |     |             W |     | W+10
    : |     |               |     |  :
    : |     |       L       |     |  :
    - *     * - - - - - - - *     *  : 
    5 |                           |  :
    - * - - * - - - - - - - * - - *  -
      : - - - - - L+20  - - - - - :

The dimensions of the pool are \(\displaystyle L\) by \(\displaystyle W.\)

The area of the pool is 1800 ft²: .\(\displaystyle LW \:=\:1800 \quad\Rightarrow\quad W \:=\:\tfrac{1800}{L}\) .[1]

The total area has dimensions \(\displaystyle L+20\) by \(\displaystyle W+10\)

We have: .\(\displaystyle A \;=\;(L+20)(W+10) \;=\;LW + 10L + 20W + 200\) .[2]


Substitute [1] into [2]: .\(\displaystyle A \;=\;L\left(\tfrac{1800}{L}\right) + 10L + 20\left(\tfrac{1800}{L}\right) + 200 \)

. . which simplifies to: .\(\displaystyle A \;=\;10L + 36000L^{-1} + 200\)


Differentiate and equate to zero: .\(\displaystyle 10 - 36000L^{-2} \:=\:0\)

Multiply by \(\displaystyle L^2\!:\;\;10L^2 - 36000 \:=\:0 \quad\Rightarrow\quad 10L^2 \:=\:36000 \quad\Rightarrow\quad L^2\:=\:3600\)

. . Hence: .\(\displaystyle L \:=\:60\)

Substitute into [1]: .\(\displaystyle W \:=\:\frac{1800}{60} \quad\Rightarrow\quad W\:=\:30\)


Therefore, the property will be: .\(\displaystyle \begin{Bmatrix}L+20 &=& 80\text{ ft} \\ & & \text{by} \\W+10 &=& 40\text{ ft} \end{Bmatrix} \)