If the semivertical angle of a cone is `45^@`. Then the rate of change of its volume is

Options

A. curved surface area times the rate of change of r
B. base area times the rate of change of l
C. base area times the rate of change of r
D. none of these

Answer

To solve the problem of finding the rate of change of the volume of a cone with a semi-vertical angle of \(45^\circ\), we can follow these steps:

### Step 1: Understand the Geometry of the Cone
Given that the semi-vertical angle of the cone is \(45^\circ\), we can establish the relationship between the height \(h\) and the radius \(r\) of the cone. Since the angle is \(45^\circ\), we have:
\[ h = r \]

### Step 2: Write the Volume Formula for the Cone
The volume \(V\) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting \(h = r\) into the volume formula, we get:
\[ V = \frac{1}{3} \pi r^2 r = \frac{1}{3} \pi r^3 \]

### Step 3: Differentiate the Volume with Respect to Time
To find the rate of change of volume with respect to time, we differentiate both sides of the volume equation with respect to time \(t\):
\[ \frac{dV}{dt} = \frac{d}{dt}\left(\frac{1}{3} \pi r^3\right) \]
Using the chain rule, we differentiate \(r^3\):
\[ \frac{dV}{dt} = \frac{1}{3} \pi \cdot 3r^2 \frac{dr}{dt} \]
This simplifies to:
\[ \frac{dV}{dt} = \pi r^2 \frac{dr}{dt} \]

### Step 4: Interpret the Result
The expression \(\frac{dV}{dt} = \pi r^2 \frac{dr}{dt}\) indicates that the rate of change of the volume of the cone is equal to the base area of the cone (\(\pi r^2\)) multiplied by the rate of change of the radius (\(\frac{dr}{dt}\)).

### Final Result
Thus, the rate of change of the volume of the cone is:
\[ \frac{dV}{dt} = \pi r^2 \frac{dr}{dt} \]