Question

The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate is its area increasing when the side of the triangle is 15 cm?

Hint:

To find the rate of change of area in such problems, differentiate the area formula with respect to time using the chain rule.

Answer 1

The area \( A \) of an equilateral triangle with side \( s \) is given by the formula: \[ A = \frac{\sqrt{3}}{4} s^2 \] We are given that \( \frac{ds}{dt} = 3 \, \text{cm/s} \) and \( s = 15 \, \text{cm} \), and we need to find \( \frac{dA}{dt} \). Differentiate both sides of the area formula with respect to time \( t \): \[ \frac{dA}{dt} = \frac{\sqrt{3}}{4} \cdot 2s \cdot \frac{ds}{dt} \] Substitute \( s = 15 \) and \( \frac{ds}{dt} = 3 \): \[ \frac{dA}{dt} = \frac{\sqrt{3}}{4} \cdot 2 \cdot 15 \cdot 3 = \frac{\sqrt{3}}{4} \cdot 90 = 75 \, \text{cm}^2/\text{s} \] Thus, the rate of change of the area is 75 cm\(^2\)/s.