SOLUTION: 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

Algebra ->  Rectangles -> SOLUTION: 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      Log On


   

Question 894730: 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 the least?
Give a geometric explanation. Be sure to include a graph with relevant points labeled.

Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
l is a bad choice for length. Choose L for length.

A is area and is constant A=4.
wL=A; perimeter p is p=2w%2B2L.
w=A%2FL allows for p=2%28A%2FL%29%2B2L.

Domain for p is L%3E0.

Knowing that A=4, p=2%284%2FL%29%2B2L
p=8%2FL%2B2L

Further,
p=8%2FL%2B2L%5E2%2FL
p=%288%2B2L%5E2%29%2FL
p=2%284%2BL%5E2%29%2FL

This is a rational function, and there is an asymptote for L=0.
L^2 becomes smaller faster than L, as L approaches 0, but looking at the separate terms 8/L and 2L^2/L, seeing 8/L increases while 2L^2/L will decrease without bound as L approaches 0. The function p of L will increase without bound as L approaches 0.

graph%28300%2C300%2C-1%2C12%2C-1%2C12%2C8%2Fx%2B2x%5E2%2Fx%29

You might try derivative and look for the local minimum, but the graph shows a minimum perimeter at about L=2.

graph%28300%2C300%2C-1%2C4%2C-1%2C12%2C8%2Fx%2B2x%5E2%2Fx%29