Question 1105794: An artist is designing a logo for a business in the shape of a circle with an inscribed rectangle.the diameter of circle is 6.5 inches and the area of the rectangle is 15 square inches.Find the dimension of the rectangle?
Found 2 solutions by Boreal, rothauserc:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! The rectangle has to be symmetric with the diameter, so the diagonals have to have diameters of 6.5 in
LW=15; L=15/W
L^2+W^2=6.5^2=42.25
225/W^2+ (W^2)=42.25 by substitution
225+W^4=42.25W^2, multiplying everything by W^2.
W^4-42.25W^2+225=0
4W^4-169W^2+900=0
W^2=(1/8)(169+/- sqrt (28561-14400); sqrt term= 119
W^2=36, W=6 using positive root, L=2.5
(6 inches x 2.5 inches) ANSWER
W^2=6.25, W=2.5 using negative root, L=6, same result.
Answer by rothauserc(4718)  (Show Source):
You can put this solution on YOUR website! Both diagonals of the rectangle have the same length as the diameter of the circle
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let l be the length of the rectangle and w be the width, then using the Pythagorean Theorem
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l^2 + w^2 = 6.5^2 = 42.25
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l = sqrt(42.25 - w^2)
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we know the area(A) of the rectangle is 15 square inches and A = l * W, then
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sqrt(42.25 - w^2) * w = 15
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square both sides of =
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(42.25 - w^2) * w^2 = 15
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42.25w^2 - w^4 = 225
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w^4 -42.25w^2 +225 = 0
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let u = w^2, then
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u^2 -42.25u + 225 = 0
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(u-36) * (u-6.25) = 0
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u = 36 or u = 6.25
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since u = w^2, then w = sqrt(u)
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w = 6 or 2.5
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we have two possible rectangles
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l = 15/6 = 2.5, w = 6
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l = 15/2.5 = 6, w = 2.5
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