SOLUTION: An oil painting is 10 inches longer than it is wide and is bordered on all sides by a 3 inch wide frame. If the area of the frame alone is 402 inches squared, what are the dimensio

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: An oil painting is 10 inches longer than it is wide and is bordered on all sides by a 3 inch wide frame. If the area of the frame alone is 402 inches squared, what are the dimensio      Log On


   

Question 1104022: An oil painting is 10 inches longer than it is wide and is bordered on all sides by a 3 inch wide frame. If the area of the frame alone is 402 inches squared, what are the dimensions of the painting?
Found 2 solutions by josmiceli, ikleyn:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +w+ = the width of the painting
+w+%2B+10+ = the length of the painting
+w%2A%28+w+%2B+10+%29+ = the area of the painting
+%28+w+%2B+6+%29%2A%28+w+%2B+10+%2B+6+%29+ = the area of painting plus frame
----------------------------------------------------------
+%28+w+%2B+6+%29%2A%28+w+%2B+16+%29+-+w%2A%28+w+%2B+10+%29+=+402+
+w%5E2+%2B+6w+%2B+16w+%2B+96+-+w%5E2+-+10w+=+402+
+12w+=+402+-+96+
+12w+=+306+
+w+=+25.5+
and
+w+%2B+10+=+35.5+ = length of the painting
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The dimensions are 25.5 x 35.5
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check:
+%28+w+%2B+6+%29%2A%28+w+%2B+16+%29+-+w%2A%28+w+%2B+10+%29+=+402+
+25.5+%2B+6+%29%2A%28+25.5+%2B+16+%29+-+25.5%2A35.5+=+402+
+31.5%2A41.5+-+905.25+-+402+
+1307.25+-+905.25+=+402+
+402+=+402+
OK

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
The problem says

L - W = 10,                     (1)
(L+2*3)*(W+2*3) - L*W = 402.    (2)


Simplify (2). It is equivalent to

(L+6)*(W+6) - LW = 402,

LW + 6L + 6W + 36 - LW = 402,

6L + 6W = 402 - 36 = 366,

6*(L+W) = 366  ====>  L + W = 366%2F6 = 61.


So, the problem is equivalent to these two equations

L - W = 10,
L + W = 61.
------------------Add the equations (both sides). You will get

2L = 10 + 61 = 71  ====>  L = 71%2F2 = 35.5.


Answer.  The dimensions of the oil painting are 35.5 inches (length) and 25.5 inches (width).