Question 224179: Please help with this question:
There are three rectangles of equal area. The length of the first rectangle is 12 inches more that its width, the length of the second rectangle is 32 inches more than its width, and the length of the third rectangle is 44 inches more than its width. If all dimensions are integers, find them.
Answer by drj(1380)   (Show Source):
You can put this solution on YOUR website! Please help with this question:
There are three rectangles of equal area. The length of the first rectangle is 12 inches more that its width, the length of the second rectangle is 32 inches more than its width, and the length of the third rectangle is 44 inches more than its width. If all dimensions are integers, find them.
Step 1. The area A of a rectangle is A=w*L where w is the width and L is the length.
Step 2. For the first rectangle, let w1 be the width and w1+12 be the length. Then A=w1(w1+12).
Step 3. For the second rectangle, let w2 be the width and w2+32 be the length.
Then A=w2(w2+32)
Step 4. For the third rectangle, let w3 be the width and w+44 be the length. Then, A=w3(w3+44)
Step 5. Since all the rectangles have equal area then
The above are equations of parabolas intersecting at point (0,-A).
The way I solve this is through trial and error where I used a spreadsheet like Microsoft Excel to find those integers where the areas are equal. The first column is the integers and then the other columns calculate the areas for each of the rectangles as a function of the integer.
Step 6. ANSWER
Dimensions for Rectangle 1 are 48 inches and 60 inches
Dimensions for Rectangle 2 are 40 inches and 72 inches
Dimensions for Rectangle 3 are 36 inches and 80 inches.
If I figured out how to do this more efficiently and analytically, I will revise my solution later. In the meantime, I'd thought you would like the answers soon.
I hope the above steps were helpful.
For free Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra or for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.
And good luck in your studies!
Respectfully,
Dr J
drjctu@gmail.com
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