An object \(1.50 \mathrm{cm}\) high is held \(3.00 \mathrm{cm}\) from a person's cornea, and its reflected image is measured to be 0.167 cm high. (a) What is the magnification? (b) Where is the image? (c) Find the radius of curvature of the convex mirror formed by the comea. (Note that this technique is used by optometrists to measure the curvature of the cornea for contact lens fitting. The instrument used is called a keratometer, or curve measurer.)

To find the magnification (m), we can use the given heights of the object and its reflected image. The magnification is the ratio of the image height (h') to the object height (h), which can be written as: \( m = \frac{h'}{h} \) We are given the object height (h = 1.50 cm) and the reflected image height (h' = 0.167 cm). Plug these values into the equation to find the magnification (m).

Once we have the magnification, we can find the image distance (d') using the relation: \( m = \frac{d'}{d} \) Where d is object distance and is given as 3.00 cm. Rearrange the equation to solve for d': \( d' = m * d \) Plug in the magnification from step 1 and the object distance to find the image distance (d').

Now, we'll use the mirror equation to find the radius of curvature (R) for the mirror formed by the cornea. The mirror equation states: \( \frac{1}{d} + \frac{1}{d'} = \frac{1}{f} \) Where f is the focal length. For a convex mirror, f = R/2. So we can rewrite the equation as: \( \frac{1}{d} + \frac{1}{d'} = \frac{2}{R} \) We have the object distance d and image distance d' from the previous steps. Rearrange the equation to solve for R: \( R = \frac{2}{\frac{1}{d} + \frac{1}{d'}} \) Plug in the values of d and d' to find the radius of curvature (R). a) Magnification (m): \( m = \frac{0.167 \, \mathrm{cm}}{1.50 \, \mathrm{cm}} \) \( m = 0.111\) b) Image distance (d'): \( d' = m * d \) \( d' = 0.111 * 3.00 \, \mathrm{cm} \) \( d' = 0.333 \, \mathrm{cm} \) c) Radius of curvature (R): \( R = \frac{2}{\frac{1}{3.00 \, \mathrm{cm}} + \frac{1}{0.333 \, \mathrm{cm}}} \) \( R = \frac{2}{0.3333 \, \mathrm{cm^{-1}} + 3.00 \, \mathrm{cm^{-1}}} \) \( R = 0.536 \, \mathrm{cm} \) So, the magnification is 0.111, the image is 0.333 cm from the cornea, and the radius of curvature of the convex mirror formed by the cornea is 0.536 cm.