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
First you need to find the area of an equilateral triangle with side x.
Base = x
Height = (√3)x/2
Area = (√3)x²/4
Thus, the area as a function of side x
A(x)=(√3)x²/4
A'(x) = (√3)x/2
Substitute x=12 cm in A'(x) to get rate of change of area (in cm²)
To find the rate at which the area of the equilateral triangle is increasing, we need to differentiate the formula for its area with respect to time.
The area of an equilateral triangle is given by the formula:
A = (√3/4) * (side length)^2
Let's call the side length of the triangle "s" and the rate at which s is increasing "ds/dt". We are given that ds/dt = √3 cm/min.
Now, let's differentiate the formula for the area of the triangle with respect to time (t):
dA/dt = d/dt [(√3/4) * s^2]
Using the power rule of differentiation, the derivative of s^2 with respect to t is 2s * ds/dt:
dA/dt = (√3/4) * 2s * ds/dt
Substituting the given values, we have:
dA/dt = (√3/4) * 2(12 cm) * (√3 cm/min)
Simplifying the expression:
dA/dt = (12/2) * (√3/2) * (√3 cm/min)
dA/dt = 6 * (3/2) * (√3 cm/min)
dA/dt = 9√3 cm^2/min
Therefore, the rate at which the area of the equilateral triangle is increasing when its edge is 12 cm long is 9√3 cm^2/min.