The side of an equilateral triangle is 'a' units and is increasing at the rate of `lambda` units/sec. The rate of increase of its area, is

Options

A. `(2)/(sqrt(3))lambda a`
B. `sqrt(3) lambda a`
C. `(sqrt(3))/(2) lambda a`
D. none of these

Answer

To find the rate of increase of the area of an equilateral triangle whose side is increasing at a given rate, we can follow these steps:

### Step 1: Understand the problem
We are given that the side of an equilateral triangle is 'a' units and is increasing at a rate of `λ` units/sec. We need to find the rate of increase of its area.

### Step 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:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

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

### Step 4: Simplify the expression
Simplifying the expression, we have:
\[ \frac{dA}{dt} = \frac{\sqrt{3}}{2} a \cdot \frac{da}{dt} \]

### Step 5: Substitute the rate of change of the side
We know from the problem that the side \( a \) is increasing at a rate of \( \frac{da}{dt} = \lambda \). Substituting this into our equation gives:
\[ \frac{dA}{dt} = \frac{\sqrt{3}}{2} a \cdot \lambda \]

### Step 6: Final expression for the rate of increase of area
Thus, the rate of increase of the area of the equilateral triangle is:
\[ \frac{dA}{dt} = \frac{\sqrt{3}}{2} a \lambda \]