The edge of a cube is decreasing at the rate of ` 0.6 cm//sec ` then the rate at which its volume is decreasing when the edge of the cube is 2cm , is

Options

A. ` 1.2 cm^(3)//sec`
B. ` 3.6 cm ^(3)//sec`
C. ` 4.8 cm ^(3)//sec`
D. ` 7.2 cm ^(3)//sec`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the problem
We need to find the rate at which the volume of a cube is decreasing when the edge of the cube is 2 cm and the edge is decreasing at a rate of 0.6 cm/sec.

### Step 2: Define the variables
Let:
- \( A \) = length of the edge of the cube (in cm)
- \( V \) = volume of the cube (in cm³)
- \( \frac{dA}{dt} \) = rate of change of the edge length (in cm/sec)
- \( \frac{dV}{dt} \) = rate of change of the volume (in cm³/sec)

Given:
- \( \frac{dA}{dt} = -0.6 \) cm/sec (negative because the edge is decreasing)
- \( A = 2 \) cm

### Step 3: Write the volume formula
The volume \( V \) of a cube is given by:
\[ V = A^3 \]

### Step 4: Differentiate the volume with respect to time
To find the rate of change of volume, we differentiate \( V \) with respect to \( t \):
\[ \frac{dV}{dt} = \frac{d}{dt}(A^3) = 3A^2 \frac{dA}{dt} \]

### Step 5: Substitute the values
Now, substitute \( A = 2 \) cm and \( \frac{dA}{dt} = -0.6 \) cm/sec into the differentiated equation:
\[ \frac{dV}{dt} = 3(2^2)(-0.6) \]

### Step 6: Calculate the values
Calculate \( 2^2 \):
\[ 2^2 = 4 \]
Now substitute this back into the equation:
\[ \frac{dV}{dt} = 3 \times 4 \times (-0.6) \]
\[ \frac{dV}{dt} = 12 \times (-0.6) = -7.2 \text{ cm}^3/\text{sec} \]

### Step 7: Interpret the result
The negative sign indicates that the volume is decreasing. Therefore, the rate at which the volume is decreasing is:
\[ \frac{dV}{dt} = 7.2 \text{ cm}^3/\text{sec} \]

### Final Answer
The rate at which the volume of the cube is decreasing when the edge is 2 cm is **7.2 cm³/sec**.

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