Chapter 8.2, Problem 56PS

Solution

To calculate : The focal length (the distance from the vertex to the focus).

The focal length is $1.3$ units.

Given information :

Scientists studying dolphin echolocation can simulate the projection of a dolphin's clicking sounds using computer models. The models originate the sounds at the focus of a parabolic reflector. The parabola in the graph models the cross section (with units in inches) of the reflector used to simulate sound projection for a bottlenose dolphin.

  

Calculation :

Given parabola opens to right hand side. So, the equation of parabola is $y^2=4px$ .

Coordinates of the vertex is $\left(3,4\right)$ that means $x=3\text{ and }y=4$ .

Substitute the values in the equation to find the value of $p$ .

  $\begin{array}{c}{y}^2=4px\\ {\left(4\right)}^2=4p\left(3\right)\\ 16=12p\\ p=1.3\end{array}$

Thus, $p=1.3$ .

Focus of parabola lie $\left|p\right|$ units from the vertex.

  $\begin{array}{c}\left|p\right|=\left|1.3\right|\\ =1.3\end{array}$

Therefore, the focal length is $1.3$ units.