Chapter 2, Problem 10CT

a.

To determine

To find: The equation that represents the cross section of the given disc.

Answer to Problem 10CT

The equation is $x^2=40y$

Explanation of Solution

Given:

Vertex at $(0,0)$ and focus at $(0,10)$

Concept Used:

The general equation of parabola that opens up is given by:

  ${(x-h)}^2=4p(y-k)$

Where, $(h,k)$ is the vertex and $p$ is the distance between the focus and vertex

Calculation:

Substitute $(h,k)=(0,0)$ in general equation, we get:

  $\begin{array}{l}$ {(x-0)}^2=4p(y-0)$=>x^2=4py\end{array}$

  $p$ is distance between $(0,0)$ and $(0,10)$ and hence $p=10-0=10$

Thus, the equation of parabola is given by:

  $\begin{array}{l}{x}^2=4(10)y\\ =>x^2=40y\end{array}$

b.

To determine

The depth of Sunfire.

Answer to Problem 10CT

The depth of the Sunfire is $8.556ft.$

Explanation of Solution

Given:

It is extended $37ft.$ on $x$ -axis and hence point on $x$ axis is $(37/2,0)=(18.5,0)$

Calculation:

The parabola is deep at $x=18.5$ and hence we need $y$ coordinate to find the depth.

Substitute $x=18.5$ in the equation of parabola

  $\begin{array}{l}$ {(18.5)}^2=40y$=>342.25=40y\\ =>\frac{342.25}{40}=y\\ =>8.556=y\end{array}$

Thus, Sunfire is $8.556$ $ft.$ deep.