Chapter 9.3, Problem 22E

(a)

To determine

To Find: The equation of a spherical buoy whose radius is 2 ft and center is $C\left(0,0,0\right)$.

Answer to Problem 22E

The equation of the spherical buoy is ${\left(x\right)}^2+{\left(y\right)}^2+{\left(z\right)}^2=4$.

Explanation of Solution

Given:

The value of radius is 2 ft and center of the spherical buoy is $C\left(0,0,0\right)$.

Calculation:

The below Figure is of the spherical buoy.

Figure (1)

The formula to evaluate the equation of a sphere whose radius is r and center is $C\left(h,k,l\right)$ is,

${\left(x-h\right)}^2+{\left(y-k\right)}^2+{\left(z-l\right)}^2=r^2$

Where,

• h, k, l are the coordinates of the center of the sphere.

Substitute 0 for h, 0 for k ,0 for l and 2 for r in the above formula.

$\begin{array}{c}{\left(x-0\right)}^2+{\left(y-0\right)}^2+{\left(z-0\right)}^2={\left(2\right)}^2\\ {\left(x\right)}^2+{\left(y\right)}^2+{\left(z\right)}^2=4\end{array}$

Thus, the equation of the spherical buoy is ${\left(x\right)}^2+{\left(y\right)}^2+{\left(z\right)}^2=4$.

(b)

To determine

To Find: The equation of the circle formed at the waterline of the buoy.

Answer to Problem 22E

The equation of the circle formed at the waterline of the buoy is ${\left(x\right)}^2+{\left(y\right)}^2=1.75$.

Explanation of Solution

Given:

The value of radius is 2 ft and center of the spherical buoy is $C\left(0,0,0\right)$. and 6 inches of the buoy is under the water.

Calculation:

The below Figure is of the spherical buoy.

Figure (1)

Trace the circle formed in the plane

$\begin{array}{c}z=-\left(2-0.5\right)\\ =-1.5\end{array}$

The equation of the spherical buoy is ${\left(x\right)}^2+{\left(y\right)}^2+{\left(z\right)}^2=4$.

Substitute $-1.5$ for z in the above equation to get the trace in $z=-1.5$ plane.

$\begin{array}{c}{\left(x\right)}^2+{\left(y\right)}^2+{\left(-1.5\right)}^2=4\\ {\left(x\right)}^2+{\left(y\right)}^2+2.25=4\\ {\left(x\right)}^2+{\left(y\right)}^2=1.75\end{array}$

Thus, the equation of the circle formed at the waterline of the buoy is ${\left(x\right)}^2+{\left(y\right)}^2=1.75$.