Question

find the dimensions of a rectangle with an area of 108 square feet that has the minimum perimeter?

Answer 1

Do you know the formula for area and perimeter?

Answer 2

a=lw and perimeter 2l+2w ?

Answer 3

yeah

Answer 4

We want to minimize perimeter, so we can use the area formula to find \(w\) in terms of \(l\).

Answer 5

Remember that they told us \(a=108\).

Answer 6

okay that's where I get lost idk whats after that

Answer 7

Okay so \[
108 = lw\\
p = 2l+2w
\]

Answer 8

Do you see how we can get rid of \(l\) by using substitution?

Answer 9

not quite how would It be done

Answer 10

\[
l = 108/w
\]

Answer 11

that's all

Answer 12

Now substitute: \[
p=2(108/w)+2w
\]

Answer 13

In order to minimize \(p\) you need to find \(dp/dw\). It will tell you which width \(p\) has max/min at.

Answer 14

You need to find \(dp/dw = 0\)

Answer 15

wio your a genius how do I find dp/dw or is it 0 then where do I plug it in

Answer 16

First you differentiate \(p\) in terms of \(w\).

Answer 17

Can you differentiate: \[
p=2(108/w)+2w
\]?

Answer 18

p-2w=2(108/w) ?

Answer 19

Find the derivative.

Answer 20

Do you know how to differentiate?

Answer 21

\[
p = 216w^{-1}+2w
\]

Answer 22

Just use power rule.

Answer 23

then would it ;be p= 216/w2+2w =0?

Answer 24

You don't know how to differentiate?

Answer 25

not really

Answer 26

\[
\frac{d(216w^{-1})}{dw} = -216w^{-2}
\]

Answer 27

\[
\frac{d(2w)}{dw} = 2
\]

Answer 28

This gives us: \[
\frac{dp}{dw} = -216w^{-2}+2
\]We are finding critical points so we let \(dp/dw=0\)\[
0= -216w^{-2}+2 \implies - 2 = -216w^{-2}\implies -2w^2=-216 \implies w^2=108
\]

Answer 29

so the dimension would be 2 and 52

Answer 30

We have a critical point when \(w=\pm \sqrt{108}\), we also have one when \(w=0\). However we know that \(w>0\), so the only critical point that matters is \(w=\sqrt{108}\).

Answer 31

Now we just need to find the length.
We go back to \(l=108/w = 108/\sqrt{108} = \sqrt{108}\).

Answer 32

The dimensions are \(w=\sqrt{108}\) and \(l=\sqrt{108}\).

Answer 33

wio you are a life saver you have time for one more ?