Related Rates IV 

Volume and Surface Area of a Sphere 

Copyright Maplesoft, a division of Waterloo Maple Inc., 2007 

Introduction 

This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.  Click on the buttons to watch the videos. 

 

The steps in the document can be repeated to solve similar problems. 

Problem Statement 

Helium is pumped into a spherical balloon at the constant rate of 25 cubic feet/minute. At what rate is the surface area of the balloon increasing at the moment when its radius is 8 feet? 

Solution 

 

Enter in the expression for the Volume of a sphere (with a radius that is a function of ) and then differentiate it to get the rate of change.

Type in the function for the Volume of a sphere with the radius set to r(t).  Right-click on the expression and choose Differentiate>t

 

(3.1)

 

 

(3.2)

 

The rate of change of volume is 25 cubic feet/minute. Solve the resulting equation for the rate of change of the radius, .

Use the equation label above   ([Ctrl][L] then equation number) to refer to the previous result, and set it equal to 25. Press [Enter]. Right-click, Solve>Isolate for>diff(r(t),t). Extract the right-hand side for later use: Right-click, Right-hand Side.

 

(3.3)

 

 

(3.4)

 

 

(3.5)

 

Enter the expression for the surface area of a sphere (with a radius that is a function of ) and then differentiate it.

Type in the function for the surface area (use the Expression palette for ) with radius r(t).  Right-click on the expression and choose Differentiate>t

 

(3.6)

 

 

(3.7)

 

Replace   with the value from step .

Copy the result above to a new line. Replace the derivative with its value by using the equation label. Press [Enter].

 

 

(3.8)

 

Replace with the given radius .

Copy the result to a new line. Replace   r(t) with 8.  Press [Enter] .

 

 

(3.9)

 

 

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