This problem will lead you through the steps to answer this question:A spherical balloon is being inflated 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 of the balloon also change. Recall that the volume of a sphere of radius rr is V=4/3(πr)^3. Note that in the setting of this problem, both V and r are changing as time t changes, and thus both V and r may be viewed as implicit functions of t, with respective derivatives dV/dt and dr/dtDifferentiate both sides of the equation V=4/3(πr)^3 with respect to t (using the chain rule on the right) to find a formula for dV/dt that depends on both r and dr/dtdV/ dt= From the above discussion, we know the value of dV/ dt= at every value of t. Next, observe that when the diameter of the balloon is 12, we know the value of the radius. In the equation dV/dt=4(πr)^2 (dr/dt) substitute these values for the relevant quantities and solve for the remaining unknown quantity, which is dr/dt. How fast is the radius changing at the instant when d=12?How fast is the radius changing at the instant when d=16?
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A spherical balloon is being inflated 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 of the balloon also change. Recall that the volume of a sphere of radius rr is V=4/3(πr)^3. Note that in the setting of this problem, both V and r are changing as time t changes, and thus both V and r may be viewed as implicit functions of t, with respective derivatives dV/dt and dr/dt
Differentiate both sides of the equation V=4/3(πr)^3 with respect to t (using the chain rule on the right) to find a formula for dV/dt that depends on both r and dr/dt
dV/ dt=
From the above discussion, we know the value of dV/ dt= at every value of t. Next, observe that when the diameter of the balloon is 12, we know the value of the radius. In the equation dV/dt=4(πr)^2 (dr/dt) substitute these values for the relevant quantities and solve for the remaining unknown quantity, which is dr/dt. How fast is the radius changing at the instant when d=12?
How fast is the radius changing at the instant when d=16?
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