Question

I normally don't have trouble with homework assignments, but this one i have no idea how to solve.

Rectangle R has varying length l and width w but with a constant perimeter of 4 feet
a) Express the area A as a function of length l. Write a list of
everything you know about this function.
b) For what values of l and w will the area of R be the greatest?
Give an algebraic explanation. Then give a geometric
explanation. Be sure to include a graph with relevant points
labeled.

Can any one help me find out how to solve this problem?

Answer 1

start by writing an equation for finding the perimeter.
can you do that ?

Answer 2

Heres my work so far
W+L=2
L=2-W And i can plug this into A=L*W
A=(2-W)*W
A=2W*W^2

But thats where i get stuck...Am i doing something wrong?
Thank you for helping me

Answer 3

the last line has a typo? * should be -

Answer 4

oops, yes, thats supposed to be -. But im still stuck on what to do next

Answer 5

if we use x for w and y for A, we can write it as
y= -x^2 +2x

that is the equation of a parabola (see attached)

Answer 6

As you can see, the vertex (where it peaks) occurs when x =1
i.e. when the width is 1

btw, for part (a) they want a function of L, and you found a function of W
you should re-do that part to get f(L)= L(2-L) = -L^2 +2L
(same parabola, so we get the same peak)

Answer 7

ohhhhh, Thats how to graph it. Thank you, So to see what values would be the greatest i would use the graph? (1,1)????

Answer 8

I expect they want you to find the vertex of the parabola
one way is to change to vertex form
y = a(x-h)^2 +k
where (h,k) is the vertex

you do that by completing the square.

Answer 9

the other way is to remember that when in standard form
y = a x^2 + bx + c

the x coordinate of the vertex is at x= -b/(2a)

Answer 10

graphing is one way to find the vertex, but to graph the parabola, people normally find the vertex using one of the ideas up above. (unless you use a computer to graph it)
to help make the graph.

Answer 11

Thank you for helping me with this! If its not to much trouble, can you help me with the second part?

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 R because of this behavior? Could you have
made a similar statement about R back in Task 1?
c) For what values of l and w will the perimeter of R be theleast? Give a geometric explanation. Be sure to include a graph with relevant points labeled.

Answer 12

how far did you get ?

Answer 13

I haven't started yet, I never got past part one. Im attempting to solve it now.

Answer 14

I think i just have to do the same thing, just for the perimeter this time. Am I right?

Answer 15

yes, it is the same approach.

Answer 16

P=2(L+W)
Area=L*W=4ft^2
W=4/L
P=2(L+4/L)
Is that right?

Answer 17

yes.

if we relabel L as x and P as y this is the same as
y = 2x +8/x

that is a hyperbola, but I'm not sure how to prove it.

If you download Geogebra
http://www.geogebra.org/cms/en/

you can make this plot

Answer 18

Ok Thanks for your help!

Answer 19

yw