If the rate of change of area of a square plate is equal to that of the rate of change of its perimeter, then length of the side, is

Options

A. 1 unit
B. 2 units
C. 3 units
D. 4 units

Answer

To solve the problem of finding the length of the side of a square plate when the rate of change of its area is equal to the rate of change of its perimeter, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
Let the length of the side of the square be denoted as \( A \). The area \( (S) \) of the square is given by:
\[ S = A^2 \]
The perimeter \( (P) \) of the square is given by:
\[ P = 4A \]

2. **Differentiate the Area**:
To find the rate of change of the area with respect to time \( t \), we differentiate \( S \):
\[ \frac{dS}{dt} = \frac{d(A^2)}{dt} = 2A \frac{dA}{dt} \]

3. **Differentiate the Perimeter**:
To find the rate of change of the perimeter with respect to time \( t \), we differentiate \( P \):
\[ \frac{dP}{dt} = \frac{d(4A)}{dt} = 4 \frac{dA}{dt} \]

4. **Set the Rates Equal**:
According to the problem, the rate of change of area is equal to the rate of change of perimeter:
\[ \frac{dS}{dt} = \frac{dP}{dt} \]
Substituting the expressions we derived:
\[ 2A \frac{dA}{dt} = 4 \frac{dA}{dt} \]

5. **Cancel Common Terms**:
Assuming \( \frac{dA}{dt} \neq 0 \) (the side length is changing), we can divide both sides by \( \frac{dA}{dt} \):
\[ 2A = 4 \]

6. **Solve for A**:
Now, we can solve for \( A \):
\[ A = \frac{4}{2} = 2 \]

7. **Conclusion**:
The length of the side of the square is \( 2 \) units.