Find an equation of the largest sphere with center $(5, 4, 8)$ that is contained in the first octant.

Neetesh Kumar | December 19, 2024

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This is the solution to Math 1C
Assignment: 12.1 Question Number 15
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Step-by-step solution:

The first octant is the region in $\mathbb{R}^3$ where $x \geq 0$, $y \geq 0$, and $z \geq 0$. For the largest sphere to be completely contained within the first octant, the sphere's surface must touch one or more coordinate planes but not extend beyond them.

The radius of the sphere is determined by the smallest distance from the center $(5, 4, 8)$ to the coordinate planes:

1. Distance to the $yz$-plane $(x = 0)$: $\text{Distance} = |x| = 5$

2. Distance to the $xz$-plane $(y = 0)$: $\text{Distance} = |y| = 4$

3. Distance to the $xy$-plane $(z = 0)$: $\text{Distance} = |z| = 8$

The smallest distance determines the largest radius that keeps the sphere within the first octant. Thus, the radius of the sphere is: $r = \min(5, 4, 8) = 4$

The general equation of a sphere with center $(h, k, l)$ and radius $r$ is: $(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$

Substituting the center $(5, 4, 8)$ and $r = 4$: $(x - 5)^2 + (y - 4)^2 + (z - 8)^2 = 4^2$

Simplify: $(x - 5)^2 + (y - 4)^2 + (z - 8)^2 = 16$

Final Answer:

The equation of the largest sphere is: $\boxed{(x - 5)^2 + (y - 4)^2 + (z - 8)^2 = 16}$

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