SOLUTION: a 5 in. by 7 in. photograph is surrounded by a frame of uniform width. The area of the frame equals the area of the photograph. Find the width of the frame

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: a 5 in. by 7 in. photograph is surrounded by a frame of uniform width. The area of the frame equals the area of the photograph. Find the width of the frame      Log On


   

Question 286261: a 5 in. by 7 in. photograph is surrounded by a frame of uniform width. The area of the frame equals the area of the photograph. Find the width of the frame
Answer by oberobic(2304) About Me  (Show Source):
You can put this solution on YOUR website!
A of picture = 5*7=35
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Frame width = x
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The area of the frame = area of the picture, which means it is 35+35 = 70
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(2x+5)(2x+7) = 70
4x^2 +14x +10x + 35 = 70
4x^2 +24x + 35 = 70
4x^2 +24x -35 = 0
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Using the quadratic solver...
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Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B24x%2B-35+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2824%29%5E2-4%2A4%2A-35=1136.

Discriminant d=1136 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-24%2B-sqrt%28+1136+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2824%29%2Bsqrt%28+1136+%29%29%2F2%5C4+=+1.21307488658818
x%5B2%5D+=+%28-%2824%29-sqrt%28+1136+%29%29%2F2%5C4+=+-7.21307488658818

Quadratic expression 4x%5E2%2B24x%2B-35 can be factored:
4x%5E2%2B24x%2B-35+=+4%28x-1.21307488658818%29%2A%28x--7.21307488658818%29
Again, the answer is: 1.21307488658818, -7.21307488658818. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B24%2Ax%2B-35+%29

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x = 1.2130 (The other root is negative, so it is spurious.)
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Checking, what is the area of a rectangle defined by the outside of the frame?
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(5+2(1.2130)) * (7 + 2(1.2130)) = (5+2.426)*(7+2.426) = 7.246*9.246 = 69.997,
which is pretty close to 70.
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Done.