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Question

my whole question didn't fit the exact question is a swimming pool has the shape of the ellipse X^2 divided by 2500 + y^2 divided by 400 equals 1, where x and y are measured in feet. cross sections perpendicular to x-axis are squares. Find the total volume of the pool. My proffessor is not very good at explaining things and i have been stuck on this for 4 hours please help.

Answer 1

This is volume by slicing. From the top view, the pool is an ellipse. The cross sections are squares, so the cross section area is the square of the values of 2y. If you plot the ellipse, you will see that it goes from -50 to 50 on the x-axis and -20 to 20 on the y-axis. Since it is symmetrical, we can take twice the integration of 0 to 50.

y = √(400(1-x²/2500))

A(x) = 4y²=1600(1-x²/2500))

V(x) = 2∫⁵⁰₀ A(x) dx =2∫⁵⁰₀ 1600(1-x²/2500)) dx = 1600[x -x³/7500]₀⁵⁰ ≈ 106666.67 ft²

Answer 2

The area A(x) of each cross section at x is the area of a square whose side length is the height function 2y(x) of the ellipse at x, so A(x) = 4y(x)². The volume is then V = ∫_–a^aA(x)dx where a = √2500 is the ellipse semi-major axis. Finally, from the ellipse equation y(x)² = b²(1 – x²/a²) where b = √400 is the ellipse semi-minor axis.

Answer 3

V = ∫_-50⁵⁰(2y)²dx = ∫_-50⁵⁰4y²dx = 8∫₀⁵⁰(1 - x²/2500)·400dx = 3200(x - x³/7500)₀⁵⁰= 320000/3 feet³