Question

Rectangle R has varying length l and width w but a constant perimeter of 4 ft.
A. Express the area A as a function of l. What do you know about this function?
B. For what values of l and w will the area of R be greatest? Give an algebraic argument. Give a geometric arguement.

Answer 1

Well, we know that: \[
2l+2w=P = 4
\]Because they give us a perimeter constraint.
This allows us to find \(w\) as a function of \(l\).
Can you start by doing that?

Answer 2

You need to solve for \(w\).

Answer 3

am I just solving for w from this equation?

Answer 4

For now, you are.

Answer 5

Okay hold on one second

Answer 6

Wait, how do you solve with two equal signs?

Answer 7

Only use one of them... in this case only use: \[
2w+2l=4
\]

Answer 8

Ok

Answer 9

Is it w=2-l ?

Answer 10

Yes

Answer 11

Now since you know that: \[
A = w\times l
\]and \(w=2-l\)
You can get \(A\) in terms of only \(l\).

Answer 12

This will finish part (a)

Answer 13

So what is \(A\) as a function of \(l\).

Answer 14

Do I substitute 2-l for w in A=wl ?

Answer 15

Yes

Answer 16

then I got \[A=2-l ^{2}\]

Answer 17

When you substitute it in, it should have parenthesis around it.

Answer 18

Meaning that you'd have to use the distributive property to simplify it.

Answer 19

\[
A = (2-l)l
\]

Answer 20

Wait, I thought I would solve it and it would become 2-l^2

Answer 21

Yeah, but your algebra is wrong.

Answer 22

Oh, nevermind...I see what I did wrong