Chapter 11.1, Problem 52E

To determine

Tofind:Thestandard equation of the sphere with center $\left(0,5,-9\right)$ and diameter 8.

Answer to Problem 52E

Thestandard equation of the sphere with center $\left(0,5,-9\right)$ and diameter 8is ${\left(x\right)}^2+{\left(y-5\right)}^2+{\left(z+9\right)}^2=16$ .

Explanation of Solution

Given information:

The center is $\left(0,5,-9\right)$ and the diameter is 8.

Calculation:

The standard equation of a sphere with center $\left(h,k,j\right)$ and radius $r$ can be calculated as follows.

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

Radius is the half of the diameter. Divide the diameter by 2 to calculate the radius.

  $\begin{array}{c}\text{Radius}r=\frac{8}{2}\\ =4\text{units}\end{array}$

Substitute $\left(h,k,j\right)=\left(0,5,-9\right)$ and 4for $r$ in the above equation to calculate the standard equation of the sphere.

  $\begin{array}{c}{\left(x-0\right)}^2+{\left(y-5\right)}^2+{\left(z-\left(-9\right)\right)}^2=4^2\\ {\left(x\right)}^2+{\left(y-5\right)}^2+{\left(z+9\right)}^2=16\end{array}$

Therefore, the standard equation of the sphere with center $\left(0,5,-9\right)$ and diameter 8 is ${\left(x\right)}^2+{\left(y-5\right)}^2+{\left(z+9\right)}^2=16$ .