The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when the side is 10 cm is

Options

A. `sqrt3` sq. unit/sec
B. 10 sq. unit/sec
C. `10sqrt3` sq. unit/sec
D. `(10)/(sqrt3)` sq. unit/sec

Explanation

To solve the problem, we need to find the rate at which the area of an equilateral triangle increases when the side length is 10 cm, given that the sides are increasing at a rate of 2 cm/sec.

### Step-by-Step Solution:

1. **Understand the Formula for the Area of an Equilateral Triangle**:
The area \( A \) of an equilateral triangle with side length \( a \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

2. **Differentiate the Area with Respect to Time**:
To find the rate of change of the area with respect to time, we differentiate the area \( A \) with respect to \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt} \left( \frac{\sqrt{3}}{4} a^2 \right) \]
Using the chain rule, we get:
\[ \frac{dA}{dt} = \frac{\sqrt{3}}{4} \cdot 2a \cdot \frac{da}{dt} = \frac{\sqrt{3}}{2} a \frac{da}{dt} \]

3. **Substitute the Given Values**:
We know from the problem that:
- The side length \( a = 10 \) cm
- The rate of change of the side length \( \frac{da}{dt} = 2 \) cm/sec

Now, substituting these values into the differentiated equation:
\[ \frac{dA}{dt} = \frac{\sqrt{3}}{2} \cdot 10 \cdot 2 \]

4. **Calculate the Rate of Change of Area**:
Simplifying the expression:
\[ \frac{dA}{dt} = \frac{\sqrt{3}}{2} \cdot 20 = 10\sqrt{3} \text{ cm}^2/\text{sec} \]

5. **Final Answer**:
The rate at which the area of the triangle increases when the side is 10 cm is:
\[ \frac{dA}{dt} = 10\sqrt{3} \text{ cm}^2/\text{sec} \]