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Chapter 23: Problem 54

In a subway station, a convex mirror allows the attendant to view activity on the platform. A woman \(1.64 \mathrm{m}\) tall is standing \(4.5 \mathrm{m}\) from the mirror. The image formed of the woman is \(0.500 \mathrm{m}\) tall. (a) What is the radius of curvature of the mirror? (b) The mirror is \(0.500 \mathrm{m}\) in diameter. If the woman's shoes appear at the bottom of the mirror, does her head appear at the top-in other words, does the image of the woman fill the mirror from top to bottom? Explain.

Short Answer

Expert verified
Question: Determine the radius of curvature of a convex mirror and whether the entire image of a woman can be seen in the mirror from top to bottom, given the following information: Height of the woman is 1.64 m, distance of the woman from the mirror is 4.5 m, and the height of the image is 0.500 m. Answer: The radius of curvature of the convex mirror is 2.160 m, and the woman's image fills the entire convex mirror from top to bottom.

Step by step solution

01

Identify the given information and required variables

We are given: - Height of the woman: \(h_o = 1.64\mathrm{m}\) - Distance of the woman from the mirror: \(d_o = 4.5\mathrm{m}\) - Height of the image: \(h_i = 0.500\mathrm{m}\) We need to find: (a) The radius of curvature of the mirror, \(R\). (b) If the woman's image fills the mirror from top to bottom.
02

Use the magnification formula to find the distance of the image from the mirror (d_i)

The magnification formula is given by: \(M = \frac{h_i}{h_o} = \frac{-d_i}{d_o}\) We can rearrange the formula and solve for \(d_i\): \(d_i =\frac{-h_i \times d_o}{h_o}\) \(d_i = \frac{-0.500 \times 4.5}{1.64}\) \(d_i = -1.365 \mathrm{m}\) The image distance \(d_i\) comes out as negative, which is correct since it confirms that this is a virtual image, as expected for a convex mirror.
03

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

The mirror formula relates the object distance (\(d_o\)), image distance (\(d_i\)), and the focal length of the mirror (\(f\)): \(\frac{1}{f} = \frac{1}{d_o} +\frac{1}{d_i}\) Rearranging the formula to find \(f\), we get: \(f = \frac{1}{\frac{1}{d_o} + \frac{1}{d_i}}\) Inserting the values for \(d_o\) and \(d_i\): \(f = \frac{1}{\frac{1}{4.5} + \frac{1}{-1.365}}\) \(f = 1.080\mathrm{m}\)
04

Find the radius of curvature of the mirror (R)

For a convex mirror, the radius of curvature R is given by: \(R = 2f\) \(R = 2 \times 1.080\) \(R = 2.160 \mathrm{m}\) The radius of curvature of the convex mirror is \(2.160 \mathrm{m}\).
05

Determine if the woman's image fills the mirror from top to bottom

The mirror's diameter is \(0.500 \mathrm{m}\). To determine whether the woman's image fills the entire mirror from top to bottom, we need to compare the size of her image (height of the image \(h_i = 0.500\mathrm{m}\)) with the mirror's height. Since the height of the image is equal to the mirror's diameter, the woman's image fills the entire convex mirror from top to bottom.

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