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(1 point) A spherical snowball is melting so that its diameter is decreasing at rate of 0.2 cm/min. At what is the rate is the volume of the snowball changing when the diameter is 8 cm? The volume is changing at a rate of cm3/min.(1 point) This problem will lead you through the steps to answer this question: Aspherical balloon is being inated at a constant rate of 20 cubic inches per second. How fast is the radius of the balloon changing at the instant the balloon's diameter is 12 inches? Is the radius changing more rapidly when d : 12 or when d : 16? Why? Draw several spheres with different radii, and observe that as volume changes, the radius, diameter, and surface area ofthe balloon also change. Recall that the volume of a sphere of radius 'r is V : 57W? Note that in the setting of this problem, both V and 'r are changing as time t changes, and thus both V and 7' may be viewed as implicit functions oft, with respective derivatives % d3 _1' and alt . Differentiate both sides of the equation if : 7: W73 with respect to 1 (using the chain rule on the right) to nd a formula for L; that depends on both 1' and if _ \"W , Cl aft 7 At this point in the problem, by differentiating we have "related the rates" of change of V and 1'. Recall that we are given in the problem that the balloon is being inated at a constant rate of 20 cubic inches per second. To which derivative does this rate correspond? 0 dr 0 K dt 0 E (it O D. None of these From the above discussion, we know the value of % at every value of t. Next, observe that when the diameter ofthe balloon is 12, we know the value of the radius. In the equation dV 2 dr .1, E : 4m- 3 substitute these values for the relevant quantities and solve for the remaining unknown quantity, which is 3' How fast is the radius changing at the instant when d : 12'? How fast is the radius changing at the instant when d : 16? Cl When is the radius changing more rapidly, when d : 12 or when d : 16'? O A. when d : 12 O B. when d : 16 O C. Neither; the rate of change of the radius is constant A road perpendicular to a highway leads to a farmhouse located 1 mile away. An automobile traveling on the highway passes through this intersection at a speed of 55mph. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 4 miles past the intersection of the highway and the road? The distance between the automobile and the farmhouse is increasing at a rate ofD miles per hour
