Chapter 9.5, Problem 60STP

a.

To determine

To solve the formula for $\frac{1}{p}$

Answer to Problem 60STP

The formula for $\frac{1}{p}$ is $\frac{q-f}{fq}$

Explanation of Solution

Given information:

The formula that can be used to determine the distance in which film should be kept from the lens to create a perfect photograph is $\frac{1}{q}=\frac{1}{f}-\frac{1}{p}$

Here $f$ represents the focal length of the lens, $q$ represents the distance from the lens to the film and $p$ represents the distance from the object to the lens

Calculation:

The formula for $\frac{1}{p}$ can be derived from the given formula as,

  $\begin{array}{l}\frac{1}{q}=\frac{1}{f}-\frac{1}{p}\\ -\frac{1}{p}=\frac{1}{q}-\frac{1}{f}\\ \frac{1}{p}=\frac{1}{f}-\frac{1}{q}\\ \frac{1}{p}=\frac{q-f}{fq}\end{array}$

Therefore, the formula for $\frac{1}{p}$ is $\frac{q-f}{fq}$

b.

To determine

To write the expression containing $f\text{ and }q$ as a single rational function

Answer to Problem 60STP

The expression containing $f\text{ and }q$ as a single rational function is $\frac{q-f}{fq}$

Explanation of Solution

Given information:

The formula that can be used to determine the distance in which film should be kept from the lens to create a perfect photograph is $\frac{1}{q}=\frac{1}{f}-\frac{1}{p}$

Here $f$ represents the focal length of the lens, $q$ represents the distance from the lens to the film and $p$ represents the distance from the object to the lens

A single rational expression is a function in which the numerator or the denominator are polynomials

The single rational expression containing $f\text{ and }q$ can be written from the help of formula of $\frac{1}{p}$ that is derived from the given formula

Therefore, the expression is $\frac{q-f}{fq}$

c.

To determine

To calculate:

The distance in which the object should be placed from the lens so that the picture will be in focus

Answer to Problem 60STP

The object should be placed at 40 centimeters from the lens then the picture will be in focus.

Explanation of Solution

Given information:

The formula that can be used to determine the distance in which film should be kept from the lens to create a perfect photograph is $\frac{1}{q}=\frac{1}{f}-\frac{1}{p}$

Here $f$ represents the focal length of the lens, $q$ represents the distance from the lens to the film and $p$ represents the distance from the object to the lens

The focal length of the lens is 8 centimeters and the lens is 10 centimeters from the film

Formula used:

  $\frac{1}{q}=\frac{1}{f}-\frac{1}{p}$

Calculation:

The distance of the object from the lens can be calculated as,

  $\begin{array}{l}\frac{1}{q}=\frac{1}{f}-\frac{1}{p}\\ \frac{1}{10}=\frac{1}{8}-\frac{1}{p}\\ \frac{1}{10}-\frac{1}{8}=-\frac{1}{p}\\ \frac{4-5}{40}=-\frac{1}{p}\\ -\frac{1}{40}=-\frac{1}{p}\\ p=40\end{array}$

Therefore, the object should be placed at 40 centimeters from the lens then the picture will be in focus.