/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 56 A camera has a telephoto lens of... [FREE SOLUTION] | 91影视

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Chapter 24: Problem 56

A camera has a telephoto lens of 240 -mm focal length. The lens can be moved in and out a distance of \(16 \mathrm{mm}\) from the film plane by rotating the lens barrel. If the lens can focus objects at infinity, what is the closest object distance that can be focused?

Short Answer

Expert verified
Answer: The closest object distance that can be focused with these specifications is approximately 3360mm.

Step by step solution

01

Find the Image Distance when focusing on the closest object

When the lens is moved inwards by 16mm, the distance from the film plane is reduced, meaning the image distance is now 240mm - 16mm: \(d_i = 240\mathrm{mm} - 16\mathrm{mm} = 224\mathrm{mm}\) With this new value for \(d_i\), we can calculate the closest object distance.
02

Use the Thin Lens Equation to Calculate the Closest Object Distance

Using the thin lens equation, \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), and the values for \(f\) and \(d_i\) from the previous step, we can find the closest object distance \(d_o\). Substituting the values, we get: \(\frac{1}{240\mathrm{mm}} = \frac{1}{d_o} + \frac{1}{224\mathrm{mm}}\) Now, we need to solve for \(d_o\).
03

Solve for Closest Object Distance

To solve for \(d_o\), first subtract \(\frac{1}{224\mathrm{mm}}\) from both sides: \(\frac{1}{240\mathrm{mm}} - \frac{1}{224\mathrm{mm}} = \frac{1}{d_o}\) Now, find a common denominator: \(\frac{224 - 240}{240 \times 224} = \frac{1}{d_o}\) Simplifying the fraction: \(\frac{-16}{240 \times 224} = \frac{1}{d_o}\) Now, take the reciprocal of the fractions: \(d_o = -\frac{240 \times 224}{16}\) Finally, since object distance is positive, we can remove the negative sign: \(d_o = \frac{240 \times 224}{16}\) Now, calculate the value for \(d_o\): \(d_o \approx 3360\mathrm{mm}\) The closest object distance that can be focused is approximately 3360mm.

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