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 the equilateral triangle is increasing, you can use the formula for the area of an equilateral triangle: A = (√3/4) * s^2, where A is the area and s is the length of each side.
First, let's find the rate at which the side length is increasing. We are given that the side length is increasing at a rate of √3 cm/min. This means that ds/dt = √3 cm/min.
Next, we differentiate the area formula with respect to time (t) using the chain rule:
dA/dt = (dA/ds) * (ds/dt)
The derivative of the area with respect to the side length (dA/ds) can be found by differentiating the area formula, which is:
dA/ds = (√3/4) * 2s
= (√3/2) * s
Now, substitute the given values of ds/dt = √3 cm/min and the length of each side s = 12 cm into the differentiation formula:
dA/dt = (√3/2) * 12 * √3
= (√3/2) * 12 * √3
= (3/2) * 12
= 18 cm^2/min
Therefore, the rate at which the area of the equilateral triangle is increasing when the edge is 12 cm long is 18 cm^2/min.