Question

Rectangle R has varying Length L and Width W but with a constant permittee of 4ft. 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 area of R be the greatest? give and algebraic and geometric explanation

Answer 1

Welcome to Openstudy

Length = L, Width = W , P = Perimeter, A = Area
2(L+ W) =P = 4; A = L*W as a function of Length, A = L*(2-L)
L + W = 2 L = 2 - W; W = 2 - L
A = 2L -L^2
when the length = width the area is greatest
A = (2-L)^2=2L - L^2
solve for L then find W
substitute in original to check

let me know if you need more help

Answer 2

so after i solve for L and then find w i will have myt final answer both of my questions?

Answer 3

Also how would you solve for L?

Answer 4

yes

(2-L)^2=2L-L^2
expand on the left hand side
(2-L)(2-L)=2L-L^2
simplify left hand side, no change on the right
2(2-L)-L(2-L)=2L-L^2
get rid of parenthesis
4-2L-2L+L^2=2L-L^2
subtract (2L-L^2) from each side
and write in standard form
2L^2-6L +4= 0
what are the factors of (2*4) that add to -6?
use -4 and -2
2L^2 -4L-2L +4 = 0
2L(L-2)-2(L-2)=0
(2L-2)(L-2)=0
2L-2 =0; L-2 =0
L=1 ; L=2
when L=1, W = 1
when L = 2, W = 0
cannot have a width of zero and therefore the only valid solution is when L=1 for the greatest area.

Answer 5

Ohhh okay. Thank you!
Is this the my final answer?

Answer 6

yes

Answer 7

you are welcome

Answer 8

Would you happen to know this 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 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 9

For the very first question that I had how would I include a graph with relevant points labeled? Like what would I graph?