The length of a rectangle is five times greater than the width, w. What is the ratio of the area of the rectangle to the perimeter of the rectangle in simplest form?

Question

1. The length of a rectangle is five times greater than the width, w. What is the ratio of the area of the rectangle to the perimeter of the rectangle in simplest form?
2. A rectangular poster has an area represented by the function . Determine the expressions for the measures of the length and width of the rectangle. What can you tell about the rectangle from your answer?
3. Robert wrote the equation, . What is the value of N for A = , B = , and C = ?

Answer 1

L=5W

A=LW = 5W^2

P= 2L+2W= 10W +2W = 12W

ratio of A to P = A/) = 5W^2/12W = 5W/12

A/P = 5W/12 = 5(L/5)/12 = L/12

A/P = L/12 is more simple, but not much more, than A/P =5W/12

4x^2 + 44x + 121 = (2x+11)^2

IF it's a square then Length=Width = 2x +11. That the polynomial is a perfect square suggests the rectangle is a square with L=W.

L=W=2x+11 means x>-11/2, x>-5.5

If the rectangle is not a square Length & Width could each be anywhere from nearly zero to nearly infinity, and anywhere in between. the larger one dimension, the smaller the other dimension. As one dimension approaches zero, the other dimension approaches infinity.

N=(A-B)/C A = (5x^2 +9x -1)/(x)(x+4) B= (3x^2 +4x +2)/(x)(x+4)

A-B = (2x^2+5x-3)/x(x+4) = (2x-1)(x+3)/(x)(x+4)

C =(-2x^2-7x -3)/x

(A-B)/C = (2x-1)(x+3)/x(x+4) times (x)(x+4)/-(2x+1)(x+3) = -1

(A-B)/C = -1

N = -1