The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm?

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the problem
We are given that the side of an equilateral triangle is increasing at a rate of 2 cm/s. We need to find the rate at which the area of the triangle is increasing when the side is 20 cm.

### Step 2: Define the variables
Let:
- \( A \) = length of the side of the equilateral triangle (in cm)
- \( \frac{dA}{dt} \) = rate of change of the side with respect to time = 2 cm/s
- \( \text{Area} = A \) (capital A denotes the area of the triangle)

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

### Step 4: Differentiate the area with respect to time
We need to find \( \frac{d(\text{Area})}{dt} \). Using the chain rule, we differentiate the area with respect to time:
\[ \frac{d(\text{Area})}{dt} = \frac{d}{dt} \left( \frac{\sqrt{3}}{4} A^2 \right) = \frac{\sqrt{3}}{4} \cdot 2A \cdot \frac{dA}{dt} \]
This simplifies to:
\[ \frac{d(\text{Area})}{dt} = \frac{\sqrt{3}}{2} A \cdot \frac{dA}{dt} \]

### Step 5: Substitute the known values
We know:
- \( A = 20 \) cm (the side length at the moment we are interested in)
- \( \frac{dA}{dt} = 2 \) cm/s (the rate of increase of the side)

Substituting these values into the equation:
\[ \frac{d(\text{Area})}{dt} = \frac{\sqrt{3}}{2} \cdot 20 \cdot 2 \]

### Step 6: Calculate the rate of change of the area
Now, calculate:
\[ \frac{d(\text{Area})}{dt} = \frac{\sqrt{3}}{2} \cdot 20 \cdot 2 = 20\sqrt{3} \text{ cm}^2/\text{s} \]

### Final Answer
The area of the triangle is increasing at a rate of \( 20\sqrt{3} \) cm²/s when the side of the triangle is 20 cm.

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