Suppose that the radius r and surface area `S=4pi r^(2)` of a sphere are differentiable functions of t. Write an equation that relates `(ds)/(dt)` to `(dr)/(dt)`.

Options

A. `(ds)/(dt)= 8 pi r (dr)/(dt)`
B. `(ds)/(dt)=4pi r (dr)/(dt)`
C. `(ds)/(dt)=4pi r^(2)(dr)/(dt)`
D. `(ds)/(dt)=(dr)/(dt)`

Answer

To find the relationship between \(\frac{ds}{dt}\) and \(\frac{dr}{dt}\), we start with the formula for the surface area \(S\) of a sphere, which is given by:

\[ S = 4\pi r^2 \]

where \(r\) is the radius of the sphere.

### Step 1: Differentiate the surface area with respect to time \(t\)

To relate \(\frac{ds}{dt}\) to \(\frac{dr}{dt}\), we need to differentiate \(S\) with respect to \(t\). Using the chain rule, we have:

\[ \frac{ds}{dt} = \frac{dS}{dr} \cdot \frac{dr}{dt} \]

### Step 2: Calculate \(\frac{dS}{dr}\)

Next, we need to find \(\frac{dS}{dr}\). Starting from the expression for \(S\):

\[ S = 4\pi r^2 \]

Differentiating \(S\) with respect to \(r\):

\[ \frac{dS}{dr} = \frac{d}{dr}(4\pi r^2) = 8\pi r \]

### Step 3: Substitute \(\frac{dS}{dr}\) back into the equation

Now, we can substitute \(\frac{dS}{dr}\) back into the equation we derived in Step 1:

\[ \frac{ds}{dt} = \frac{dS}{dr} \cdot \frac{dr}{dt} = 8\pi r \cdot \frac{dr}{dt} \]

### Final Equation

Thus, the equation that relates \(\frac{ds}{dt}\) to \(\frac{dr}{dt}\) is:

\[ \frac{ds}{dt} = 8\pi r \cdot \frac{dr}{dt} \]