As viewed from above, a swimming pool has the shape of the ellipse given by

(x^2/3600)+(y^2/1600)= 1
The cross sections perpendicular to the ground and parallel to the y-axis are squares. Find the total volume of the pool.

To find the total volume of the pool, we need to integrate the cross-sectional areas along the y-axis and then multiply by the length of the pool.

The equation of the ellipse is given by (x^2/3600) + (y^2/1600) = 1.

Let's find the equation of one of the cross-sectional squares parallel to the y-axis. We can set y as a constant value and solve for x.

For a given value of y, the equation becomes:
(x^2/3600) + (y^2/1600) = 1

Rearranging the equation:
x^2/3600 = 1 - (y^2/1600)
x^2 = 3600 - (3600/1600) y^2
x^2 = 3600(1 - (9/4) y^2/400)
x^2 = 3600(1 - (9/4) (y^2/400))
x^2 = 3600(1 - (9/4) (y/20)^2)

This equation represents the sides of the square cross-section for a given value of y.

The length of each side of the square is given by 2 times the absolute value of x.

To find the cross-sectional area, we square the length of each side:
Area = (2|x|)^2
Area = 4x^2

Now, we need to integrate the cross-sectional area along the y-axis from -20 to 20 (since the shape is symmetric).

The total volume of the pool can be found by integrating the cross-sectional areas and multiplying by the length of the pool, which is 40.

Let's integrate the cross-sectional area along the y-axis and then multiply by the length:

Total volume = 40 ∫[from -20 to 20] 4x^2 dy

= 40 ∫[from -20 to 20] 4 * 3600(1 - (9/4) (y/20)^2) dy

Now, we can simplify and solve this integral to find the total volume of the pool.

To find the total volume of the swimming pool, we need to integrate the area of each cross section perpendicular to the ground along the y-axis.

Given that the cross sections are squares, we can find the side length of each square by taking the absolute value of the difference between the positive and negative y-intercepts of the ellipse.

The equation of the ellipse is given by (x^2/3600) + (y^2/1600) = 1.

To find the y-intercepts, we substitute x = 0 into the equation and solve for y.

(0^2/3600) + (y^2/1600) = 1
(y^2/1600) = 1
y^2 = 1600
y = ±40

Taking the absolute value of the difference between the y-intercepts, we get a side length of 40 - (-40) = 80 for each square cross section.

To find the volume, we integrate the area of each cross section along the y-axis from -40 to 40.

The area of each square cross section is (side length)^2 = 80^2 = 6400.

Thus, the volume is given by integrating the area function over the y-axis:

V = ∫[from -40 to 40] 6400 dy
V = 6400 ∫[from -40 to 40] dy
V = 6400 [y] [from -40 to 40]
V = 6400 (40 - (-40))
V = 6400 (80)
V = 512,000 cubic units

Therefore, the total volume of the swimming pool is 512,000 cubic units.

Integrating along x, the area of each cross-section is (2y)^2 = 4y^2.

The equation of the ellipse means

4y^2 = 6400 - 6400x^2/3600 = 6400 - 8/9 x^2

v = ∫[-60,60] 6400 - 8/9 x^2 dx
let 'er rip.

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