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Chapter 23: Problem 48

In her job as a dental hygienist, Kathryn uses a concave mirror to see the back of her patient's teeth. When the mirror is \(1.20 \mathrm{cm}\) from a tooth, the image is upright and 3.00 times as large as the tooth. What are the focal length and radius of curvature of the mirror?

Short Answer

Expert verified
Answer: The focal length of the concave mirror is 0.40 cm, and the radius of curvature is 0.80 cm.

Step by step solution

01

Write down the given information

We are given the following information: - Object distance (distance between the tooth and the mirror), \(v = 1.20 \, \mathrm{cm}\) - Magnification, \(m = 3.00\)
02

Write the mirror formula and magnification formula for a concave mirror

The mirror formula relates the object distance (\(v\)), the image distance (\(u\)), and the focal length (\(f\)) of a concave mirror: \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] The magnification formula for a concave mirror relates the magnification (\(m\)) to the object distance (\(u\)) and the image distance (\(v\)) as: \[ m = \frac{v}{u} \]
03

Use the magnification formula to find the object distance (\(u\))

We have the magnification \(m = 3.00\) and the image distance \(v = 1.20\, \mathrm{cm}\). Using the magnification formula, we can find the object distance (\(u\)): \[ u = \frac{v}{m} = \frac{1.20\, \mathrm{cm}}{3.00} = 0.40\, \mathrm{cm} \]
04

Use the mirror formula to find the focal length (\(f\))

Now that we have both the object distance (\(u = 0.40 \, \mathrm{cm}\)) and the image distance (\(v = 1.20 \, \mathrm{cm}\)), we can use the mirror formula to find the focal length (\(f\)): \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} = \frac{1}{0.40\, \mathrm{cm}} + \frac{1}{1.20\, \mathrm{cm}} \] Find their common denominator and add the fractions: \[ \frac{1}{f} = \frac{3}{1.20\, \mathrm{cm}} \] Now, find the reciprocal to get the focal length (\(f\)): \[ f = \frac{1.20 \,\mathrm{cm}}{3} = 0.40\, \mathrm{cm} \]
05

Find the radius of curvature (\(R\)) of the mirror

The radius of curvature (\(R\)) of a concave mirror is twice its focal length. Therefore, we can find the radius of curvature by multiplying the focal length (\(f = 0.40 \,\mathrm{cm}\)) by 2: \[ R = 2f = 2(0.40\, \mathrm{cm}) = 0.80\, \mathrm{cm} \] So, the focal length of the concave mirror is \(0.40\, \mathrm{cm}\), and the radius of curvature is \(0.80\, \mathrm{cm}\).

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