Question

As viewed from above, a swimming pool has the shape of the ellipse x21600+y2400=1, where x and y are measured in feet. The cross sections perpendicular to the x-axis are squares. Find the total volume of the pool.

Answer 1

To calculate the volume of the elliptical swimming pool, integrate the area of square cross sections derived from the ellipse's equation along the major axis from -40 to 40 feet.

Step-by-step explanation:

Finding the Volume of the Elliptical Swimming Pool

To find the total volume of an elliptical swimming pool with the given equation x^2/1600 + y^2/400 = 1, we need to understand its geometric properties. This equation represents an ellipse, and, as per the question, the cross sections perpendicular to the x-axis are squares. This implies that for any given x-value within the ellipse, the length of the side of a cross-sectional square is twice the y-value of that ellipse.

The volume of this type of three-dimensional object can be computed by integrating the area of the square cross-sections along the x-axis. The area A of a square is given by A=s^2, where s is the side length. Here, s=2y, so A=4y^2. Since y^2 can be expressed in terms of x using the equation of the ellipse, we have y^2 = 400(1-x^2/1600). Substituting this into the area formula gives A = 4(400)(1-x^2/1600).

Integrating this area along the length of the x-axis from -40 to 40 feet (the major axis length of the ellipse), will give us the total volume V. The definite integral \int_{-40}^{40} 1600(1-x^2/1600) dx will yield the volume of the pool.

Answer 2

I suppose the ellipse should have the equation

`(x^2)/(1600)+(y^2)/(400)=1\implies y=\pm\frac{√(1600-x^2)}4`

Then the cross sections are squares with side lengths

`\frac{√(1600-x^2)}4-\left(-\frac{√(1600-x^2)}4\right)=\frac{√(1600-x^2)}2`

so that each section contributes an area of

`\left(\frac{√(1600-x^2)}2\right)^2=\frac{1600-x^2}4`

Then the total volume of the pool would be given by

`\displaystyle\int_(-40)^(40)\frac{1600-x^2}4\,\mathrm dx=\boxed{\frac{64,000}3}`