Chapter 8.7, Problem 1MRPS

Solution

(a)

To find : An equation in standard from for the cross section of each dish if the vertex of one dish is at the origin and the vertex of the other dish is on the positive $x-$ axis.

The equation of the dish with vertex at origin is $y^2=6x$ and the equation of the dish with vertex on the positive $x-$ axis is $y^2=182\left(x-47\right)$ .

Given information :

A person at the focus of one of two facing parabolic dishes can here even a very soft sound made at the focus of the other dish. Two such "whisper dishes" are positioned with their vertices $47$ feet apart. Each dish's focus is 1.5 feet from its vertex.

  

Formula used:

The standard form of equation of a parabola with vertex at $\left(0,0\right)$ and focus at $\left(p,0\right)$ is given by:

  $y^2=4px$

The standard form of the equation of a parabola with vertex $\left(h,k\right)$ and focus at $\left(p,0\right)$ is:

  ${\left(y-k\right)}^2=4p\left(x-h\right)$

The distance $\left(d\right)$ between two points $\left(x_1,y_1\right)\text{and}\left(x_2,y_2\right)$ is given by:

  $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Explanation :

The vertex of one dish is at origin that is $\left(0,0\right)$ and the focus is 1.5 feet from its vertex. From the figure it is evident that the dishes are parabolas with vertical axis and focus at $\left(p,0\right)$ .

To find the equation of the dish with vertex at origin, substitute $p=1.5$ .

  $\begin{array}{c}{y}^2=4px\\ =4\left(1.5\right)x\\ =6x\end{array}$

The vertex of the other dish is on the positive $x-$ axis, so let the vertex be $\left(x,0\right)$ . The distance between two dishes is $47$ feet. So,

  $\begin{array}{c}47=\sqrt{{\left(x-0\right)}^2+{\left(0-0\right)}^2}\\ 47=\sqrt{x^2}\\ x=47\end{array}$

So, the vertex of the other dish is at $\left(47,0\right)$ . The focus of the dish is 1.5 feet from its vertex. Let the focus be $\left(p,0\right)$ , so

  $\begin{array}{c}1.5=\sqrt{{\left(47-p\right)}^2+{\left(0-0\right)}^2}\\ 1.5=47-p\\ p=45.5\end{array}$

To find the equation of the dish with vertex on the positive $x-$ axis, substitute $k=0,h=47$ .

  $\begin{array}{c}{\left(y-0\right)}^2=4\left(45.5\right)\left(x-47\right)\\ y^2=182\left(x-47\right)\end{array}$

Therefore, the equation of the dish with vertex at origin is $y^2=6x$ and the equation of the dish with vertex on the positive $x-$ axis is $y^2=182\left(x-47\right)$ .

(b)

To calculate : The depth of each dish if the diameter of each dish is $67$ inches.

The depth of the dish with vertex at origin is $1.3\text{ feet}$ and the depth of the dish with vertex on the positive $x-$ axis is $47.04\text{ feet}$ .

Given information :

A person at the focus of one of two facing parabolic dishes can here even a very soft sound made at the focus of the other dish. Two such "whisper dishes" are positioned with their vertices $47$ feet apart. Each dish's focus is 1.5 feet from its vertex.

  

Calculation :

The depth of each dish if the diameter of each dish is $67$ inches or $5.6$ feet.

The dish with vertex at origin extends $\frac{5.6}{2}=2.8$ feet to the either side of the vertex, so substitute $y=2.8$ in $y^2=6x$ .

  $\begin{array}{c}{\left(2.8\right)}^2=6x\\ x=\frac{7.84}{6}\\ \approx 1.3\end{array}$

Now, substitute $y=2.8$ in $y^2=182\left(x-47\right)$ .

  $\begin{array}{c}{\left(2.8\right)}^2=182\left(x-47\right)\\ 7.84=182\left(x-47\right)\\ x=\frac{7.84}{182}+47\\ \approx 47.04\end{array}$

So, the vertex of the other dish is at $\left(47,0\right)$ . The focus of the dish is 1.5 feet from its vertex. Let the focus be $\left(p,0\right)$ , so

  $\begin{array}{c}1.5=\sqrt{{\left(47-p\right)}^2+{\left(0-0\right)}^2}\\ 1.5=47-p\\ p=45.5\end{array}$

To find the equation of the dish with vertex on the positive $x-$ axis, substitute $k=0,h=47$ .

  $\begin{array}{c}{\left(y-0\right)}^2=4\left(45.5\right)\left(x-47\right)\\ y^2=182\left(x-47\right)\end{array}$

Therefore, the depth of the dish with vertex at origin is $1.3\text{ feet}$ and the depth of the dish with vertex on the positive $x-$ axis is $47.04\text{ feet}$ .

(c)

To calculate : The distance between the focus and vertex for a dish that is $8$ feet in diameter and is $1.25$ feet deep.

The distance between the focus and vertex of the given dish is $3.2\text{ feet}$ .

Given information :

A person at the focus of one of two facing parabolic dishes can here even a very soft sound made at the focus of the other dish. Two such "whisper dishes" are positioned with their vertices $47$ feet apart. Each dish's focus is 1.5 feet from its vertex.

  

Calculation :

The dish that is $8$ feet in diameter, so

  $\begin{array}{c}y=\frac{8}{2}\\ =4\end{array}$

The dish is $1.25$ feet deep, so

  $x=1.25$

To calculate the distance between the focus and vertex, substitute $x=1.25$ and $y=4$ in $y^2=4px$ .

  $\begin{array}{c}{\left(4\right)}^2=4p\left(1.25\right)\\ 16=5p\\ p=3.2\end{array}$

Therefore, the distance between the focus and vertex of the given dish is $3.2\text{ feet}$ .