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 area from one end of the pool to the other.
Since the cross sections perpendicular to the ground are squares, we can find the side length of the squares at each point on the ellipse.
Given the equation of the ellipse: (x^2)/3600 + (y^2)/1600 = 1
Let's find the side length of the square at a point (x, y) on the ellipse.
Since the cross section is perpendicular to the ground, the side length of the square is equal to 2y.
So, the cross-sectional area A at a particular point (x, y) is given by A = (2y)^2 = 4y^2.
To find the total volume, we need to integrate the cross-sectional areas along the y-axis.
Let's find the limits of integration for y.
From the equation of the ellipse, we have (x^2)/3600 + (y^2)/1600 = 1.
If we isolate y, we get y = ±sqrt(1600 - (x^2)/3600).
Since the ellipse is symmetric along the y-axis, we only need to integrate from y = 0 to y = sqrt(1600 - (x^2)/3600), and then multiply the result by 2.
Now, let's set up the integral for the total volume V.
V = 2 * ∫[from y = 0 to sqrt(1600 - (x^2)/3600)] (4y^2) dy
To simplify the integral, we can pull out the constant 4 and integrate y^2.
V = 8 * ∫[from y = 0 to sqrt(1600 - (x^2)/3600)] y^2 dy
Now, we can evaluate this integral.
Integrating y^2, we get (1/3) * y^3.
V = 8 * [(1/3) * y^3] evaluated from y = 0 to sqrt(1600 - (x^2)/3600)
V = 8 * [(1/3) * (sqrt(1600 - (x^2)/3600))^3 - (1/3) * 0^3]
V = 8 * [(1/3) * (1600 - (x^2)/3600)^(3/2)]
Now, integrate this expression with respect to x from where the ellipse intersects the x-axis, which corresponds to x = ±60.
To find the total volume, we need to integrate the expression V = 8 * [(1/3) * (1600 - (x^2)/3600)^(3/2)] from x = -60 to 60.
Therefore, the total volume of the pool is:
V = 8 * ∫[from x = -60 to 60] [(1/3) * (1600 - (x^2)/3600)^(3/2)] dx.
Evaluating this integral will give us the final answer.