Question

This time rectangle R has varying length l and width w but with a constant area of 4 square feet.

a) Express the perimeter P as a function of length l. What type of function is P? What is the domain of P?

b) Describe the asymptotic behavior of P. What can you say about rectangle Rbecause of this behavior? Could you have made a similar statement about Rback in Task 1?

c) For what values of land w will the perimeter of R be the least? Give a geometric explanation. Be sure to include a graph with relevant points labeled.

Answer 1

a) Write formula for the perimeter in terms of l and w

Answer 2

How do you do that?
I'm really confused

Answer 3

Perimeter = total length of the boundary.
Now write the formula for perimeter.

Answer 4

Like p= 4 times ( L + W)?

Answer 5

p = l+l+w+w = 2l+2w = 2(l+w)
So, p = 2times (l+w)

Answer 6

Now area = const
Area of a rectangle = l*w
so l*w = 4 ft^2
w = 4/l units

Answer 7

Put this w in the formula for perimeter you'll get p as function of l

Answer 8

Is all of this for section a ?

Answer 9

yes

Answer 10

So how do we find the domain?

Answer 11

What is the form of p in l ?

Answer 12

I got p= 2?

Answer 13

Since p = 2*(l+w) and w = 4/l ,so
p = 2*(l +4/l)

Answer 14

Check what values l can hold for a meaningful p.

Answer 15

This is where I'm at ..
How do I find the domain?

Answer 16

Think. What if l =0 ?

Answer 17

Well if L was zero everything would equal zero wouldn't it?

Answer 18

No. check it again.

Answer 19

Am I plugging it to p= 2 (l+4/l)?

Answer 20

yup

Answer 21

But 4/l is undefined because you can't divide 4/0

Answer 22

Right! Good going. Length of a rectangle being zero has no meaning. Can you think about any other value of l which will have no meaning either logically or plugging it in the formula for p

Answer 23

Or would my Answer be 8??

Answer 24

Don't mind my last message haha I was just overthinking things

Answer 25

4/0 is zero is undefined. so p is undefined.
Could think of any other l for which p is undefined

Answer 26

So is my domain that? That's it's undefined ?

Answer 27

Nope

Answer 28

Domain means values of l for which p is defined.

Answer 29

So whatnot my domain?

Answer 30

What is my domain? Zero?

Answer 31

Ok. l can take any value except 0. So the domain is all real numbers except zero. You can write it like this \[l \in R - \left\{ 0 \right\}\]

Answer 32

ohh okay thank you!
and What can you say about rectangle R because of this behavior?

Answer 33

For what values of l and w will the perimeter of R be the least? Give a geometric explanation. Be sure to include a graph with relevant points labeled
how do I graph with relevant points labeled?

Answer 34

what type of function is P?