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The sides of an equilateral triangle are...

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?

A

`10cm^(2)//s`

B

`sqrt(3)cm^(2)//s`

C

`10sqrt(3)cm^(2)//s`

D

`10/3cm^(2)//s`

Text Solution

AI Generated Solution

The correct Answer is:
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. ---

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: \[ ...
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Knowledge Check

  • 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

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