Rate of Change

Volume of Inflating Spherical Balloon as a Function of Time

A meteorologist is inflating a spherical balloon with a helium gas. If the radius of a balloon is changing at a rate of 1.5 cm/sec., express the volume V of the balloon as a function of time t (in seconds). Hint: Use composite function relationship Vsphere = 4/3 πr3 as a function of x (radius), and x (radius) as a function of t (time).

A.   V(t) = 5/2 πt3 C.   V(t) = 9/2 πt3
B.   V(t) = 7/2 πt3 D.   V(t) = 3/2 πt3


Rate of Change of Volume of Sand in Conical Shape

A conveyor is dispersing sands which forms into a conical pile whose height is approximately 4/3 of its base radius. Determine how fast the volume of the conical sand is changing when the radius of the base is 3 feet, if the rate of change of the radius is 3 inches per minute.

A.   2π ft/min C.   3π ft/min
B.   4π ft/min D.   5π ft/min


Rate of change of surface area of sphere

Gas is escaping from a spherical balloon at the rate of 2 cm3/min. Find the rate at which the surface area is decreasing, in cm2/min, when the radius is 8 cm..

52-53 Two cars traveling from the same point but going to different directions

Problem 52
A car drives south at 20 mi/hr. Another car, starting from the same point at the same time and traveling 40 mi/hr, goes east for 30 minutes then turns north. Find the rate of rotation of the line joining the cars (a) 1 hour after the start; (b) at the time the second car makes its turn.

Maxima and Minima | Applications

Graph of the Function y = f(x)
The graph of a function y = f(x) may be plotted using Differential Calculus. Consider the graph shown below.



As x increases, the curve rises if the slope is positive, as of arc AB; it falls if the slope is negative, as of arc BC.

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