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Rate of Change of Surface Area on an Expanding Sphere

Robert J. Lopez

Maplesoft Fellow

Introduction

An example of a related-rates problem in differential calculus that asks for the rate of change of surface area on a sphere whose volume expands at a constant rate is solve here via the syntax-free paradigm in Maple. A statement of the problem is as follows.

Helium is pumped into a spherical balloon at the constant rate of 25 cu ft per min.

At what rate is the surface area of the balloon increasing at the moment when its radius is 8 ft?

Recently, after presenting the solution in a Maplesoft Webinar, I was asked if it were possible to see an animation for this process. So, after a quick presentation of a solution, this worksheet will try to answer the request for an animation. Of course, we first have to consider just what is it that is to be displayed in the animation. It's easy enough to show an expanding sphere, but the question of real interest is the varying rate of change of surface area. How is the change in surface area to be visualized, let alone animated?

Traditional Solution

Analysis

Given

Find

Solution

Define the functions and

•	Write Context Menu: Assign Function

•	Write Context Menu: Assign Function

Solve the equation for

•	Write the equation Press the Enter key.

•	Context Menu: Solve_Isolate Expression for_diff(r(t),t)

(1)

In , replace with its value from the equation

•	Expression palette: Evaluation template Evaluate at from the equation

•	Context Menu: Evaluate and Display Inline

•	Make the replacement .

eval(diff(S(t), t), diff(r(t), t) = (25/4)/(Pi*r(t)^2)) =

Further Investigations

If , then , so . Moreover, , which implies and r(t) = (75*t/(4*Pi))^(1/3) . The consequences of these results are summarized in Table 1.




Table 1 Relationships between , and

Graphics

Figure 1 shows the linear behavior in (in red) and the growth of (in green). It was surprising to see the two graphs cross, but upon thought, this is to be expected for any constant value of . (The equation always has solution , so in this example, .)

Figure 2 shows (black) which increases because it is and which decreases because it is .

Figure 1 Graph of (red) and (green)

Figure 2 Graph of (black) and (green)

Figure 3 animates the expanding sphere, and graphs . The pixel density of the viewing screen limits the accuracy of the slider, so that it might not be possible to obtain exactly.