The height of an equilateral triangle is increasing at a rate of 3 cm/min. How fast is the area changing when his 5 cm? Give your answer correct to two decimal places.

Answer 1

So we have a triangle with all three sides equal, and each angle equal to 60 degrees.

The formula for the area A given side s is : A = (√3/4) * s²

The height h is a perpendicular bisector of one of the angles, and it forms a right angle with the opposite side.

From trigonometry we have h/s = sin 60 = √3/2, so s = 2h/√3

For convenience we rewrite the area formula in terms of h. We get A = h²/√3

Take the derivative of both sides using implicit differentiation:

dA/dt = (2/√3)*h* dh/dt

Finally plugging in h is 5 and dh/dt is 3, we get dA/dt = 10 * √3, which is around 17.32 cm²/min

(watch the units)

Answer 2

Let x = length of each side and Area = A = (1/2)(base)(height).

Base = x and Height = h = (√3/2)x. So, A = (√3/4)x².

Therefore, dA/dt = (√3/2)(x)(dx/dt)

When h = 5, x = (2/√3)(5) = 10 / √3. Also, we are given that dh/dt = 3.

So, dA/dt = (√3/2)(10/√3)(3) = 15.00 cm²/min