As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 ± 0.02 cm and a thickness of 0.050 ± 0.005 cm. (a) Find the average volume of a cookie and the uncertainty in the volume. (b) Find the ratio of the diameter to the thickness and the uncertainty in this ratio.

Solution 59P Step 1 The volume of a cylindrical disc is given by V = r d2 Where r is the radius of the disc and t is the thickness. In this given problem we have d (8.50±0.02 cm) r = 2 = 2 = 4.25 ± 0.01 cm And t = 0.050 ± 0.005 cm So the volume is 4 2 2 3 V = r3t = (4.25 cm) (0.050 cm) = 2.84 cm Step 2 Now the relative error in radius is r 0.01 r = 4.25= .002 And relative error in thickness is t= 0.005 c= 0.1 t 0.050 cm So the relative error in the volume is V = 2 r + t= 2(0.02) + 0.1 = 0.14 V r t Hence absolute error in volume is V = V × (0.14) = (2.84 cm )(0.14) = 0.4 cm 3 Hence the volume with its uncertainty can be reported as V = 2.8 ± 0.4 cm 3