Find a parametric representation for the surface: the part of the sphere $x^2 + y^2 + z^2 = 64$ that lies above the cone $z = \sqrt{x^2 + y^2}$.

(Enter your answer as a comma-separated list of equations. Let $x$, $y$, and $z$ be in terms of $u$ and/or $v$.)

Neetesh Kumar | November 14, 2024

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This is the solution to Math 1D
Assignment: 16.6 Question Number 15
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Step-by-step solution:

The given sphere equation is: $x^2 + y^2 + z^2 = 64$

This represents a sphere with radius $8$ (since $\sqrt{64} = 8$) centered at the origin.

The cone equation is: $z = \sqrt{x^2 + y^2}$

This cone opens upward from the origin at a $45^\circ$ angle. We need the part of the sphere that lies above this cone, which corresponds to points on the sphere where $z \geq \sqrt{x^2 + y^2}$.

Step 1: Set up the Parametric Form for the Sphere

For a sphere, a standard parametric representation in spherical coordinates is:

• $x = R \sin(\theta) \cos(\phi)$
• $y = R \sin(\theta) \sin(\phi)$
• $z = R \cos(\theta)$

where $R = 8$, $\theta$ is the polar angle (from $0$ to $\pi$), and $\phi$ is the azimuthal angle (from $0$ to $2\pi$).

So, for our sphere of radius $8$, the parametric equations are: $x = 8 \sin(\theta) \cos(\phi)$ $y = 8 \sin(\theta) \sin(\phi)$ $z = 8 \cos(\theta)$

Step 2: Apply the Condition to Lie Above the Cone

The condition for points to be above the cone $z = \sqrt{x^2 + y^2}$ on the sphere translates to: $z \geq \sqrt{x^2 + y^2}$

Substituting the parametric forms, we get: $8 \cos(\theta) \geq \sqrt{(8 \sin(\theta) \cos(\phi))^2 + (8 \sin(\theta) \sin(\phi))^2}$

Simplifying inside the square root: $8 \cos(\theta) \geq \sqrt{64 \sin^2(\theta) (\cos^2(\phi) + \sin^2(\phi))}$ Since $\cos^2(\phi) + \sin^2(\phi) = 1$, this reduces to: $8 \cos(\theta) \geq 8 \sin(\theta)$ $\cos(\theta) \geq \sin(\theta)$

Dividing both sides by $\cos(\theta)$ (assuming $\theta \neq \frac{\pi}{2}$): $1 \geq \tan(\theta)$ This implies: $\theta \leq \frac{\pi}{4}$

Step 3: Final Parametric Equations

Therefore, the parametric representation for the part of the sphere that lies above the cone is: $x = 8 \sin(\theta) \cos(\phi)$ $y = 8 \sin(\theta) \sin(\phi)$ $z = 8 \cos(\theta)$

where:

• $0 \leq \theta \leq \frac{\pi}{4}$
• $0 \leq \phi < 2\pi$

Final Answer:

The parametric representation is:

$r(x, y, z) = \bigg(8 \sin(\theta) \cos(\phi), 8 \sin(\theta) \sin(\phi), 8 \cos(\theta)\bigg)$

with $0 \leq \theta \leq \frac{\pi}{4}$ and $0 \leq \phi < 2\pi$.

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