# Grr (optimization word problem)

• Nov 21st 2006, 08:33 PM
Nitz456
Grr (optimization word problem)
"A rectangular swimming pool is to be built with an area of 1800 sq. feet. The owner wants 5-foot wide decks along either side, and 10-foot wide decks at both ends. Find the dimensions of the smallest piece of property on which the pool can be built satisfying these conditions."

help:(
• Nov 22nd 2006, 03:55 AM
earboth
Quote:

Originally Posted by Nitz456
"A rectangular swimming pool is to be built with an area of 1800 sq. feet. The owner wants 5-foot wide decks along either side, and 10-foot wide decks at both ends. Find the dimensions of the smallest piece of property on which the pool can be built satisfying these conditions."

Hello,

1. I've attached a sketch of the described swimming pool.

2. $A_{water}=l\ \cdot \ w=1800$. Thus: $l=\frac{1800}{w}$

3. The area including the decks is:

$A=(l+20)\ \cdot \ (w+10)$. Now plug in the value of l:

4. $A(w)=(\frac{1800}{w}+20)\ \cdot \ (w+10)$. Expand the RHS of this equation:

$A(w)=1800+\frac{18000}{w}+20w+200=2000+\frac{18000 }{w}+20w$

5. This function has an extreme value (maximum or minimum, you are interested in the minimum) if the first derivative equals zero:

$A'(w)=-\frac{18000}{w^2}+20$
$0=-\frac{18000}{w^2}+20$. Solve for w:

I've got: w = 30. Plug in this value to calculate l. I've got l = 60.

EB
• Nov 22nd 2006, 03:57 AM
earboth
Quote:

Originally Posted by Nitz456
"A rectangular swimming pool ...

Hello,

it's me again. I forgot to attach the sketch. Here it is:

EB