The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. How far is the area increasing when the side is 10 cms?

Options

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

Explanation

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

### Step 1: Understand the problem
We are given that the sides of an equilateral triangle are increasing at a rate of 2 cm/sec. We need to find out how fast the area of the triangle is increasing when the side length is 10 cm.

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

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

### Step 4: Substitute the known values
We know that:
- \( a = 10 \) cm (the side length at the moment we are interested in)
- \( \frac{da}{dt} = 2 \) cm/sec (the rate at which the side length is increasing)

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

### Step 5: Calculate \( \frac{dA}{dt} \)
Now we can simplify:
\[ \frac{dA}{dt} = \frac{\sqrt{3}}{2} \cdot 20 = 10\sqrt{3} \text{ cm}^2/\text{sec} \]

### Final Answer
Thus, the area of the equilateral triangle is increasing at a rate of \( 10\sqrt{3} \) cm²/sec when the side length is 10 cm.

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