It is your first day at work as a summer intern at an optics company. Your supervisor hands you a diverging lens and asks you to measure its focal length. You know that with a converging lens, you can measure the focal length by placing an object a distance $s$ to the left of the lens, far enough from the lens for the image to be real, and viewing the image on a screen that is to the right of the lens. By adjusting the position of the screen until the image is in sharp focus, you can determine the image distance $s^{\prime}$ and then use Eq. (34.16) to calculate the focal length $f$ of the lens. But this procedure won't work with a diverging lens - by itself, a diverging lens produces only virtual images, which can't be projected onto a screen. Therefore, to determine the focal length of a diverging lens, you do the following: First you take a converging lens and measure that, for an object $20.0 \mathrm{~cm}$ to the left of the lens, the image is $29.7 \mathrm{~cm}$ to the right of the lens. You then place a diverging lens $20.0 \mathrm{~cm}$ to the right of the converging lens and measure the final image to be $42.8 \mathrm{~cm}$ to the right of the converging lens. Suspecting some inaccuracy in measurement, you repeat the lenscombination measurement with the same object distance for the converging lens but with the diverging lens $25.0 \mathrm{~cm}$ to the right of the converging lens. You measure the final image to be $31.6 \mathrm{~cm}$ to the right of the converging lens. (a) Use both lens-combination measurements to calculate the focal length of the diverging lens. Take as your best experimental value for the focal length the average of the two values. (b) Which position of the diverging lens, $20.0 \mathrm{~cm}$ to the right or $25.0 \mathrm{~cm}$ to the right of the converging lens, gives the tallest image?