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Volume Calculus Question Please Help.?
As viewed from above, a swimming pool has the shape of the ellipse
x^2/2500+y^2/900=1
The cross sections perpendicular to the ground and parallel to the y-axis are squares. Find the total volume of the pool. (Assume the units of length and area are feet and square feet respectively. Do not put units in your answer.) Volume=?
1 Answer
- ?Lv 79 years agoFavourite answer
x^2/2500+y^2/900=1
ellipse
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x²/a² + y²/b² = 1
horizontal major axis; a > b
x²/2500 + y²/900 = 1
a = ±√2500 = ±50
interval [-50,50] along the x- axis
x²/2500 + y²/900 = 1
y²/900 = 1 - x²/2500
y² = 900 - 9x² / 25
y = ±√( 900 - 9x² / 25 )
y₁ = √( 900 - 9x² / 25 ) and y₂ = -√( 900 - 9x² / 25 )
width = y₁ - y₂ = 2√( 900 - 9x² / 25 )
so cross-sectional square has sides = 2√( 900 - 9x² / 25 )
thickness of the square = Δx
volume of each square = (width)² Δx
total volume using interval [-50, 50]
`````````````` ```````````````` ````````````````
V = ∫(width)² dx
V = ∫(2√( 900 - 9x² / 25 ))² dx
V = ∫ 4( 900 - 9x² / 25 ) dx
V = ∫ ( 3600 - 36x² / 25 ) dx
V = 3600x - 12x³ / 25
with [-50,50]
V = 240000
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