Rates of Change

Question

In the triangle ABC, AB=2x, AC= x+1 and ∠A =150 degrees.Show that the area of the triangle is 0.5x(x+1). If x is increasing at a rate of 2 cm s^-1,find the rate at which the area is increasing when x=4cm.

Answer 1

Area = 1/2bh.  The height, h, of the triangle can be drawn outside the triangle, so that it is perpendicular to the extension of AC. Height = 1/2(2x), because it's a 30-60-90 triangle, so the opposite side (h) = 1/2 hypotenuse (2x).

 

So, since the base = (x + 1), then area, A = 1/2x(x + 1)

 

If we differentiate the area formula wrt time, t, we get:

dA/dt = 1/2[dx/dt(x +1) + xdx/dt]  (product rule)

 

since dx/dt = 2, and x =4, we get: dA/dt = 1/2[(2)(4 +1) + (4)(2)]

 

so dA/dt = 9 cm²/s