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

Options

A. `10 cm^(2) // sec`
B. `10 sqrt(3) cm^(2) // sec `
C. ` (10)/( 3) cm^(2) // sec`
D. `sqrt( 3) cm^(2) // sec`

Answer

To solve the problem, we need to find the rate at which the area of an equilateral triangle is increasing 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 Problem**:
We are given that the side of an equilateral triangle is increasing at a rate of \( \frac{dA}{dt} = 2 \) cm/sec. We need to find the rate of change of the area of the triangle when the side length \( A = 10 \) cm.

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

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

4. **Substituting Known Values**:
We know \( \frac{da}{dt} = 2 \) cm/sec and \( a = 10 \) cm. Substituting these values into the equation:
\[ \frac{dA}{dt} = \frac{\sqrt{3}}{2} \cdot 10 \cdot 2 \]
Simplifying this gives:
\[ \frac{dA}{dt} = 10\sqrt{3} \text{ cm}^2/\text{sec} \]

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