Chapter 8.2, Problem 59PS

Solution

(a)

To calculate : The width of the beam of light from the searchlight's reflector.

The width of the beam is $20\text{ inches}\text{.}$

Given information :

Searchlights use parabolic reflectors to project their beams. The cross section of a $9.5$ inch deep searchlight reflector has equation $x^2=10.5y$ .

Calculation :

Consider the equation $x^2=10.5y$ .

To find the width of the beam, substitute $y=9.5$ .

  $\begin{array}{c}{x}^2=10.5\left(9.5\right)\\ x^2=99.75\\ x=\sqrt{99.75}\\ x=9.99\end{array}$

This means the beam extends $9.99$ inches to both sides of the vertex $\left(0,0\right)$ . So, the width of the beam will be,

  $\begin{array}{c}\text{Width}\text{of}\text{beam}=2\times 9.99\\ \approx 20\text{ inches}\text{.}\end{array}$

Therefore, the width of the beam is $20\text{ inches}\text{.}$

(b)

To determine : The equation for cross section of a reflector that has same depth as original reflector but projects a wider beam and the width of the beam of the reflector.

The equation for cross section of a reflector that has same depth as original reflector but projects a wider beam is $x^2=50y$ and the width of the beam of the reflector is $43.6\text{ inches}\text{.}$

Given information :

Searchlights use parabolic reflectors to project their beams. The cross section of a $9.5$ inch deep searchlight reflector has equation $x^2=10.5y$ .

Explanation :

Consider the given equation $x^2=10.5y$ .

It is in the form of $x^2=4py$ .

The width of the beam will be wider than the present one if $4p>10.5$ .

So, substitute $4p=50$ . Now, the equation will be $x^2=50y$ and to find the width of the new beam, substitute $y=9.5$ .

  $\begin{array}{c}{x}^2=50\left(9.5\right)\\ x^2=475\\ x=\sqrt{475}\\ x=21.79\end{array}$

This means the beam extends $21.79$ inches to both sides of the vertex $\left(0,0\right)$ . So, the width of the beam will be,

  $\begin{array}{c}\text{Extended}\text{width}\text{of}\text{the}\text{beam}=2\times 21.79\\ \approx 43.6\text{ inches}\text{.}\end{array}$

Therefore, the equation for cross section of a reflector that has same depth as original reflector but projects a wider beam is $x^2=50y$ and the width of the beam of the reflector is $43.6\text{ inches}\text{.}$

(c)

To determine : The equation for cross section of a reflector that has same depth as original reflector but projects a narrower beam and the width of the beam of the reflector.

The equation for cross section of a reflector that has same depth as original reflector but projects a narrower beam is $x^2=8y$ and the width of the beam of the reflector is $17.4\text{ inches}$ .

Given information :

Searchlights use parabolic reflectors to project their beams. The cross section of a $9.5$ inch deep searchlight reflector has equation $x^2=10.5y$ .

Explanation :

Consider the given equation $x^2=10.5y$ .

It is in the form of $x^2=4py$ .

The width of the beam will be narrower than the present one if $4p<10.5$ .

So, substitute $4p=8$ . Now, the equation will be $x^2=8y$ and to find the width of the new beam, substitute $y=9.5$ .

  $\begin{array}{c}{x}^2=8\left(9.5\right)\\ x^2=76\\ x=\sqrt{76}\\ x=8.72\end{array}$

This means the beam extends $8.72$ inches to both sides of the vertex $\left(0,0\right)$ . So, the width of the beam will be,

  $\begin{array}{c}\text{Width}=2\times 8.72\\ \approx 17.4\text{ inches}\end{array}$

Therefore, the equation for cross section of a reflector that has same depth as original reflector but projects a wider beam is $x^2=8y$ and the width of the beam of the reflector is $17.4\text{ inches}$ .