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# help

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A painting with dimensions 10 inches by 14 inches is placed in a picture frame (of constant width), increasing its area to 221 square inches. How many inches is the width of the picture frame?

Jul 23, 2019

#1
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x = width of frame

original height 10    new height   10 + 2x

original width   14    new width    14 + 2x

and

(10+ 2x)(14+2x) = 221

4x^2+48x -81 = 0       Use quadratic Formula to find  x = 1.5 inch width of frame

Jul 23, 2019
#2
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Let  x  be the width of the border......then one dimension of the picture and frame  =  ( 10 + 2x)

And the other dimension of the picture and frame  = ( 14 + 2x)

So....the area  =

(10 + 2x) ( 14 + 2x)  = 221      simplify

140 + 28x + 20x + 4x^2  = 221

4x^2 + 48x + 140  = 221       subtract 221 from both sides

4x^2 + 48x  - 81  = 0       factor

(2x - 3) (2x+ 27)  = 0        set each factor to 0   and solve for x

(2x - 3)  = 0                         2x  + 27  = 0

2x  = 3                                  2x  = -27

x  = 3/2  = 1.5 inches            x  = -27/2  = - 13.5  reject

So.....the width is 1.5 inches   Jul 23, 2019