Question

An apple modeled by taking all the points in and on a sphere of radius 8 in. is cored with a vertical cylinder of radius 2 in. Use inequalities in cylindrical coordinates to describe the set of all points that remain in the apple once the core is removed. (Use symbolic notation and fractions where needed.)

Answer 1

{p, Φ, θ; 2 ≤ p ≤ 8; 0 ≤ Φ ≤ 2π, 0 ≤ θ ≤ 2π}.

Step-by-step explanation:

Recall that y = p sinΦsinθ, x = p sinΦcosθ and z = p cosΦ ( from spherical coordinate system). Also, p = √(x^2 + y^2 + z^2), cosΦ = z/p and θ = tan^-1 (y/x).

So, from the question we are given sphere of radius 8in that is the inequality representation will now be;

x^2 + y^2 + z^2 ≤ 8^2 -----------------(1).

The next thing to do is to slot in the equation of x, y and z into (1) above.

(p sinΦcosθ)^2 + (p sinΦsinθ)^2 + (p cosΦ)^2 ≤ 64 --------------------------------(2).

After solving and factorizing Equation (2) above, we then have the equation (3) below;

p^2 sin^2 Φ(cos^2 θ + sin^2 θ) + p^2 cos^2 Φ ≤ 64 ---------------------------------(3).

[ Recall that cos^2 θ + sin^2 θ= 1].

p^2 sin^2 Φ+ p^2 cos^2 Φ ≤ 64.

p^2 (sin^2 + cos^2 Φ) ≤ 64.

p^2 ≤ 64.

p ≤ 8.

We are also given that the vertical cylinder of radius 2 in. That is 2 ≤ p ≤ 8.

Also, for a sphere the angle of Φ and θ ranges from 0 - 2π.

Therefore, the final answer;

p, Φ, θ; 2 ≤ p ≤ 8; 0 ≤ Φ ≤ 2π, 0 ≤ θ ≤ 2π.