Each side of an equilateral triangle is increasing at a rate of √3 cm/min. find the rate at which its area increasing when its edge is 12 cm long
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To find the rate at which the area of an equilateral triangle is increasing, we can use the formula for the area of an equilateral triangle: Area = (√3 / 4) * s^2, where "s" is the length of each side of the triangle.
Given that each side is increasing at a rate of √3 cm/min and the edge is 12 cm long, we can find the rate at which the area is increasing by taking the derivative of the area formula with respect to time.
Differentiating the area formula with respect to time, we get:
dA/dt = d/dt [(√3 / 4) * s^2]
= (√3 / 4) * 2s * ds/dt (by the chain rule)
Substituting s = 12 cm and ds/dt = √3 cm/min, we have:
dA/dt = (√3 / 4) * 2 * 12 * √3
= (√3 / 4) * 24 * √3
= 6 * 3
= 18 cm^2/min
Therefore, when the edge of the equilateral triangle is 12 cm long and each side is increasing at a rate of √3 cm/min, the area of the triangle is increasing at a rate of 18 cm^2/min.