Chapter 11.1, Problem 72E

a.

To determine

Standard equation of the sphere.

Answer to Problem 72E

  $x^2+y^2+z^2=16,000,000$

Explanation of Solution

Given information:

Centre: $C(0,0,0)$

Radius: $4000$

Formula used:

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

We identify $h,j,k$ from the centre’s coordinates.

Standard equation of the sphere is:

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

We identify $h,j,k$ from the centre’s coordinates.

Now finding the equation of this sphere:

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

  $\begin{array}{l}C(0,0,0)=\left(h,j,k\right)\\ h=0\\ j=0\\ k=0\end{array}$

  $\begin{array}{l}{\left(x-0\right)}^2+{\left(y-0\right)}^2+{\left(z-0\right)}^2={\left(4000\right)}^2\\ \\ \boxed{x^2+y^2+z^2=16,000,000}\end{array}$

b.

To determine

What trace could represent lines of longitude that run north-south.

Explanation of Solution

Given information:

Centre: $C(0,0,0)$

Radius: $4000$

The line of longitude that run north-south contain the diameter passing through $z-axis$ and have the same radius (4000 miles). Since we considered theEarth as a sphere, they have the shape of circle.

c.

To determine

What trace could represent lines of latitude that run east-west.

Explanation of Solution

Given information:

Centre: $C(0,0,0)$

Radius: $4000$

The line of latitude that run east-west contain could be represented by $xy-traces$ . Since we considered the Earth as a sphere, they have the shape of circle. These circles are parallel to each other. The greatest circle is the Equator which has the radius $0000$ miles.