# Minimum total area

• February 21st 2009, 01:52 PM
jbray
Minimum total area
A garden is being designed to have a planted area of 12m^2. Around the edge there will be a path, 1m wide on two opposite sides and 1.5m wide on the other two opposite sides. Find the dimentions of the garden so that minimum total area of the garden and path is used.

• February 21st 2009, 04:30 PM
TheEmptySet
Quote:

Originally Posted by jbray
A garden is being designed to have a planted area of 12m^2. Around the edge there will be a path, 1m wide on two opposite sides and 1.5m wide on the other two opposite sides. Find the dimentions of the garden so that minimum total area of the garden and path is used.

Draw a diagram (not included) it always helps with word problems.

let l be the length and w be the width.

we know that $12=lw \iff w=\frac{12}{l}$ be we also know that that area we are trying to minimize is A=(l+2)(w+3) (if you don't see why draw the diagram)

Now subbing the first into the 2nd we get

$A=(l+2)\left(\frac{12}{l}+3\right)=12+3l+\frac{24} {l}+6=18+3l+\frac{24}{l}$

Just take the derivative from here.

Good luck