Question 1112670: a parabolic trough 10 meters long 4 meters wide cross the top and 3 meters deep is filled with water at depth of 2 meters. Find the volume of the water in the through
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! a parabolic trough 10 meters long 4 meters wide cross the top and 3 meters deep is filled with water at depth of 2 meters. Find the volume of the water in the through
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Find the equation of the parabola.
Make the x-axis the top of the trough.
--> y = ax^2 - 3 and the 2 points (-2,0) and (2,0)
0 = a*4 - 3
a = 3/4
--> y = 3x^2/4 -3 is the parabola.
Find the intersections of the parabola and the line y = -1.
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-1 = 3x^2/4 - 3
x^2 = 8/3
x = +/- sqrt(24)/3
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Translating the parabola up 1 unit --> the zeroes are x = +/- sqrt(24)/3 and a 3rd point is (0,-2)
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y = ax^2 - 2
0 = a*24/9 - 2
a = 2*(9/24)
a = 3/4
--> y = 3x^2/4 - 2
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INT(3x^2/4 -2) = x^3/4 - 2x + C
Find area from x = 0 to x = sqrt(24)/3
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Area of one side = (sqrt(24)/3)^3 - 2sqrt(24)/3 sq meters
= (24/9)*sqrt(24)/3 - 2sqrt(24)/3
= 8sqrt(24)/9 - 2sqrt(24)/3
= 4sqrt(6)/9
2 times --> 8sqrt(6)/9 sq meters area of the parabola
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Vol = area*length = 80sqrt(6)/9 cubic meters
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