The side of a square sheet of metal is increasing at 3 centimetres per minute. At what rate is the area increasing when the side is 10 cm long ?

Answer

To solve the problem, we need to find the rate at which the area of a square is increasing when the side of the square is 10 cm long, given that the side is increasing at a rate of 3 cm per minute.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( s \) be the length of the side of the square (in cm).
- The area \( A \) of the square is given by the formula:
\[ A = s^2 \]

2. **Differentiate the Area with Respect to Time**:
- We need to find the rate of change of the area with respect to time, \( \frac{dA}{dt} \). To do this, we differentiate the area formula with respect to time \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(s^2) = 2s \frac{ds}{dt} \]

3. **Substitute Known Values**:
- We know that \( \frac{ds}{dt} = 3 \) cm/min (the rate at which the side is increasing).
- We need to find \( \frac{dA}{dt} \) when \( s = 10 \) cm.
- Substitute \( s = 10 \) cm and \( \frac{ds}{dt} = 3 \) cm/min into the differentiated equation:
\[ \frac{dA}{dt} = 2(10) \cdot (3) \]

4. **Calculate the Rate of Change of Area**:
- Now, calculate \( \frac{dA}{dt} \):
\[ \frac{dA}{dt} = 2 \cdot 10 \cdot 3 = 60 \text{ cm}^2/\text{min} \]

5. **Conclusion**:
- The area of the square is increasing at a rate of 60 cm² per minute when the side is 10 cm long.

### Final Answer:
The area is increasing at a rate of **60 cm²/min** when the side of the square is 10 cm long.