PLEASE HELP SOLVE FOR X

Question

 My family room is rectangular and it’s 18 feet long with a width 1¼ times the length of the study. The area of my family room is 2½ times the area of the study. What is the length of the study

Answer 1

Given:

L_f = 18 ft

W_f = 1.25 L_s

A_f = 2.5 A_s

Assume:

The widths of both the Study and the Family Room are colinear and lie on top of each other and is shared between the Family Room and the Study (i.e. two rectangles up against each other with that shared side length)

Find:

Ls = ? ft

Solution:

A_f = L_f * W_f = 18 * 1.25 L_s A_f = 22.5 L_s

A_s = L_s * W_s = L_s * 1.25 L_s A_s = 1.25 L_s²

22.5 L_s = 2.5 * 1.25 L_s² = 3.125 L_s²

180 L_s = 25 L_s² (DO NOT DIVIDE BY L_s!!)

25 L_s² - 180 L_s = 0

L_s * ( 25 L_s - 180 ) = 0

Two solutions exist

"Either"

L_s= 0 ft which it cannot be because that is an illogical house design and won't result in measurable areas in a house.

Or

25 L_s - 180 = 0

25 L_s = 180

∴ L_s = 7.2 ft

I hope this helps! Message me in the comments if you have any questions, comments, or concerns!