Q13. - Algebra 1

Question

When a person does not have a clear vision, an optometrist can prescribe lenses to correct the condition. The power $P$ of a lens, in a unit called diopters, is equal to 1 divided by the focal length f, in meters, of the lens.

a. Graph the inverse variation $P=\frac{1}{f}$.

b. Find the powers of lenses with focal lengths $+0.2$ to $-0.4$ meters.

Step-by-Step Solution

Answer

a. Graph of the inverse variation $P=\frac{1}{f}$:

 

b. The power of a lens with $+0.2$ focal length is $+5$ diopters and the power of a lens with $-0.4$ focal length is $-2.5$ diopters.

Part a. Step 1. State the concept.

When $y$ varies inversely as $x$ then it is known as an inverse variation.

Part a. Step 2. State the form of inverse variation.

Relationship of the form $xy=k$ or $y=\frac{k}{x}$ where $x,y\ne 0$ for some nonzero constant $k$ is known as inverse variation. Here $P=\frac{1}{f}$ that is., $Pf=1$, clearly when compared with $xy=k$ it can be observed that the equation is an inverse variation with $k=1$.

Part a. Step 3. Plot the graph.

The graph of inverse variation $P=\frac{1}{f}$ is given by:

Part b. Step 1. State the concept.

Relationship of the form $xy=k$ or $y=\frac{k}{x}$ where $x,y\ne 0$ for some nonzero constant $k$ is known as inverse variation i.e., $y$ varies inversely as $x$.

Part b. Step 2. Calculate the P when the focal length is + 0 . 2 .

Substitute 0.2 for $f$ into the formula $P=\frac{1}{f}$ and simplify for $P$.

$$\begin{gathered} P=\frac{1}{f} \\ =\frac{1}{0.2} \\ =\frac{10}{2} \\ =5 \end{gathered}$$

Part b. Step 3. Calculate the P when the focal length is - 0 . 4 .

Substitute $-0.4$ for f into the formula $P=\frac{1}{f}$ and simplify for $P$.

$P=\frac{1}{f}$

   $=-\frac{1}{0.4}$

   $=-\frac{10}{4}$

   $=-2.5$

Therefore, the power of the lens with $+0.2$ focal length is $+5$ diopters and the power of the lens with $-0.4$ focal length is $-2.5$ diopters.