Question

A circular pond with radius 1 metre and a maximum depth of 1 metre has the shape of a paraboloid, so that its depth 2 is 2 = x²+y2-1. What is the total volume of the pond? How does this compare with the case where the pond has the same radius but has the shape of a hemisphere?

Answer 1

To find the total volume of the pond, we integrate the function representing the shape of the paraboloid using polar coordinates. The volume of the pond can be found by evaluating the integral over the circular region. The volume of the hemisphere with the same radius can be directly calculated using the formula for the volume of a hemisphere.

Step-by-step explanation:

To find the total volume of the circular pond, we need to integrate the function representing the shape of the paraboloid. The function is given by z = x^2 + y^2 - 1, where z represents the depth of the pond and (x, y) represents the coordinates on the surface of the pond. Since the pond is circular with radius 1, we can evaluate the integral over the circular region using polar coordinates. The integral is ∫∫(r^2 - 1)rdrdθ, where r ranges from 0 to 1 and θ ranges from 0 to 2π. Solving this integral will give us the total volume of the pond.

Comparing with a hemisphere, the volume of a hemisphere with radius 1 is given by V = (2/3)πr^3. So we can directly calculate the volume of the hemisphere with the given radius of 1.