Question

The surface area of a sphere of radius r is S = 4πr². Its volume is

V = 4πr³/3.

a. Use MuPAD to find the expression for dS/dV.

b. A spherical balloon expands as air is pumped into it. What is the rate of increase in the balloon’s surface area with volume when its volume is 30 in.³?

Answer 1

The equations given are

S = 4πr²

V = 4/3 πr³

 

We need to find the value dS/dV using MATLAB. We can find this using the differentiation rule.

dS/dV = (dS/dr)/(dV/dr)

 

(a)

The MATLAB code is given below:

Input:

syms r

S = 4*pi*r^2;

V = (4/3)*pi*r^3;

dif = diff(S, r)/diff(V, r)

Output:

 

We see that

dS/dV = 2/r

 

(b)

Now, volume is V = 30

V = 30

⇒ r/3πr³ = 30

⇒ r = 1.9275

 

Hence,

dS/dV = 2/r = 2/1.9275 = 1.0375.

Hence,

dS/dV = 2/r = 2/1.9275 = 1.0375.