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You are making a frame of uniform width for a picture that is to be displayed at the local museum.
The picture is 3.25 feet tall and 3 feet wide.
The museum has allocated 15 square feet of wall space to display the picture.
What should the width be of the frame in order to us all of the allocated space?"
Let x = the width of the frame
If you draw this (a rectangle inside a rectangle) you will see the overall dimensions will be twice the width of the frame plus the picture dimension
then the over all dimensions will be
(3.25+2x) by (3+2x)
The overall area equation
(2x+3.25)(2x+3) = 15
4x^2 + 6x + 6.5x + 9.75 = 15
4x^2 + 12.5x + 9.75 - 15 = 0
A quadratic equation
4x^2 + 12.5x - 5.25 = 0
we have to use the quadratic formula here
in this equation: a=4, b=12.5, c=-5.25
Two solutions, but we only want the positive one
x = .375 ft, which would be .375(12) = 4.5 inches is the width of the frame
Confirm this by finding the area (2*.375 = .75)
(3.25+.75)*(3+.75) = 15 sq/ft