Math Help - Grr (optimization word problem)

1. 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

2. 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

3. Originally Posted by Nitz456
"A rectangular swimming pool ...
Hello,

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

EB