Let's explore some consequences of the angularsize formula. (a) Suppose you are asked to calculate an object's apparent, or angular size, when given its actual size and distance. Rearranging the formula 5 = r x a to solve for a yields one of the following equations. Even if you are unsure of your algebra skills, ask yourself this simple question. Which equation agrees with both of the following commonsense statements? . The closer an object is (small r), the larger it will appear (large a). . The larger an object is (large 5), the larger it will appear (large a). a=$xr l' 3:, S 03:2 r (b) Juanita's ngertip is 1 centimeter wide and her arm is 60 cm long. What angle, in radians, does her ngertip span in her eld of view? (Use the formula you chose in the previous step.) rad (c) Now let's convert that angle to degrees. There are about 57 degrees in one radian. 0 Since most people have a similar ratio of ngertip width to arm length, most people will block roughly ---Se|ect--- a in their own eld of view, when holding their finger in front of their face at arm's length. Challenge! The planet Mars has a diameter of approximately 6,780 kilometers (5 = 6.78 x 103 km). When it is nearly as close to Earth as it ever gets, it lies about 54,600,000 km away (r = 5.46 x 107 km). The human eye can resolve details down to an angular limit of one arc minute (1'). Can humans resolve Mars without optical aid? (d) Calculate the angular diameter of Mars, in radians, when closest to Earth. re d (e) Convert this angular or apparent size to units of arc minutes and compare your answer with the resolving limit of the human eye. arc minutes