Q35. - Algebra 2

Question

The reflective surface in a flashlight has parabolic cross section that can be modelled by the equation $y=\frac{1}{3}{x}^2$, where x and y are in centi-meters. How far from the vertex the filament bulb should be located?

Step-by-Step Solution

Answer

The filament of the bulb should be located $0.75\text{ cm}$ away from the focus.

Step 1. Write down the given information.

The reflective surface in a flashlight has parabolic cross section that can be modelled by the equation $y=\frac{1}{3}{x}^2$, where x and y are in centi-meters.

Step 2. Concept used.

For two different forms of equations of parabola stated below, use the following key-concept to find vertex, axis of symmetry, focus, directrix, direction of opening of parabola and length of latus rectum.

 $\boxed{\begin{array}{ccc}\text{Form of equations}& y=a{\left(x-h\right)}^2+k& x=a{\left(y-k\right)}^2+h\\ \text{Vertex}& \left(h,k\right)& \left(h,k\right)\\ \text{Axis of symmetry}& x=h& y=k\\ \text{Focus}& \left(h,k+\frac{1}{4a}\right)& \left(h+\frac{1}{4a},k\right)\\ \text{Directrix}& y=k-\frac{1}{4a}& x=h-\frac{1}{4a}\\ \text{Direction of opening}& \begin{array}{l}\text{upward if }a>0,\\ \text{downward if }a<0\end{array}& \begin{array}{l}\text{right if }a>0,\\ \text{left if }a<0\end{array}\\ \text{Length of latus rectum}& \left|\frac{1}{a}\right|\text{units}& \left|\frac{1}{a}\right|\text{units}\end{array}}$

Step 3. Calculation.

In order to find the how far from the vertex should the filament bulb should be located, use the equation for focus of parabola from the equation.

$\text{Focus}=\left(h,k+\frac{1}{4a}\right)$

Since, the vertex for the equation $y=\frac{1}{3}{x}^2$ is $\left(0,0\right)$. Therefore, $\left(h,k\right)=\left(0,0\right)\text{ and }a=\frac{1}{3}$.

Therefore, focus is evaluated as:

 $\begin{array}{c}\left(h,k+\frac{1}{4a}\right)=\left(0,0+\frac{1}{4\left(\frac{1}{3}\right)}\right)\\ =\left(0,0.75\right)\end{array}$

Hence the filament of the bulb should be located $0.75 cm$away from the focus.

Step 4. Conclusion.

The filament of the bulb should be located  $0.75\text{ cm}$ away from the focus.