SOLUTION: Suppose a rectangular movie screen has an area of 10416 square feet. Find the dimensions of the screen if it is 19 feet longer than it is wide.
THANKS!
Question 114393: Suppose a rectangular movie screen has an area of 10416 square feet. Find the dimensions of the screen if it is 19 feet longer than it is wide.
THANKS! Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! First let L represent the length of the movie screen and let W represent the width of the screen.
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You are told that the length is 19 feet longer than the width. So you can say in equation form
that:
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L = W + 19
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You are also told that the area of the screen is 10416 sq ft. But the area of a rectangle
such as this screen is is found by multiplying its length L by its width W. So we can write
in equation form that:
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Area = L * W
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Substitute 10416 for A and W + 19 for L and you have:
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10416 = (W + 19)*W
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Multiply out the right side by multiplying W times each of the terms in the parentheses
and the equation becomes:
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10416 = W^2 + 19W
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Get this into standard quadratic form by subtracting 10416 from both sides to make the
equation:
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0 = W^2 + 19W - 10416
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Just to get it into a more conventional form, transpose the equation (switch sides) to
make it:
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W^2 + 19W - 10416 = 0
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This is in the standard quadratic form of:
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aW^2 + bW + c = 0
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and by comparing this standard form with the conventional for for this problem, we can see
that a = 1, b = 19, and c = -10416
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For the standard quadratic form, according to the quadratic formula the answers for the
unknown are:
.
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Substitute 1 for a, 19 for b, and -10416 for c and you have:
.
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Solve for the term under the radical. and . Add
these two terms to get that the term under the radical is .
This reduces the answer to:
.
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Taking the square root of 42025 on a calculator we get 205. Substituting this result further
reduces the equation for W to:
.
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Multiply out the denominator ... 2*1 = 2 and this simplifies the equation to:
.
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The numerator must be positive so that the value of W is positive. [It would make no sense
to have a negative width.] Therefore, we can disregard the negative sign preceding the
205 and the equation becomes just:
.
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The two terms in the numerator combine to 186 and the equation is simplified to:
.
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This tells you that the small dimension of the screen is 93 feet. The larger dimension
(the length) is 19 feet more or 93 + 19 = 112 ft.
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So the screen is 93 by 112 feet.
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Check ... the area of the screen is 93 * 112 = 10416 square feet, just as the problem
says it should be. Our answer checks.
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Hope this helps you to understand the problem and how to use the quadratic formula in
solving it.
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