Samantha puts her face \(32.0 \mathrm{cm}\) from a makeup mirror and notices that her image is magnified by 1.80 times. (a) What kind of mirror is this? (b) Where is her face relative to the radius of curvature or focal length? (c) What is the radius of curvature of the mirror?

Recall the mirror equation for a mirror: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. The magnification of a mirror is given by the formula: \(m = \frac{d_i}{d_o}\), where \(m\) is the magnification, \(d_i\) is the image distance, and \(d_o\) is the object distance.

Given that Samantha's face is magnified, we can deduce that the mirror must be a concave mirror, as only concave mirrors can produce magnified images.

We are given the magnification \(m\) and the object distance \(d_o\). Use the magnification formula to calculate the image distance \(d_i\). \(m = \frac{d_i}{d_o}\) Plug in the given values, \(m = 1.80\) and \(d_o = 32.0 \mathrm{cm}\). \(1.80 = \frac{d_i}{32.0}\) Multiply both sides by \(32.0\) to get \(d_i\): \(d_i = 1.80\cdot 32.0 = 57.6\mathrm{cm}\) The image distance is \(57.6 \mathrm{cm}\).

The focal point is located between the mirror and the center of curvature (radius of curvature), and since the image distance is greater than the object distance and magnified, Samantha's face (the object) is between the focal point and the mirror's surface. Thus, her face is closer to the mirror than to the radius of curvature/focal length.

We have the object distance \(d_o\) and the image distance \(d_i\), but we need to find the focal length \(f\) first. Use the mirror equation to do this: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\) Plug in the values \(d_o = 32.0 \mathrm{cm}\) and \(d_i = 57.6 \mathrm{cm}\): \(\frac{1}{f} = \frac{1}{32.0} + \frac{1}{57.6}\) The common denominator of \(32\) and \(57.6\) is \(144\), so the equation becomes: \(\frac{1}{f} = \frac{4.5 + 2.5}{144}\) Simplify the expression on the right-hand side: \(\frac{1}{f} = \frac{7}{144}\) To find \(f\), take the reciprocal of both sides: \(f = \frac{1}{\frac{1}{f}} = \frac{144}{7} = 20.57\mathrm{cm}\) Since the mirror is concave, the focal length should be negative: \(f = -20.57\mathrm{cm}\) Now, we can find the radius of curvature \(R\) for a concave mirror, which is twice the focal length: \(R = 2f = 2(-20.57) = -41.14\mathrm{cm}\) The radius of curvature of the mirror is \(-41.14 \mathrm{cm}\).