Question

the included angle of the two sides of constant equal length s of an isosceles triangle is (theta)
a) show that the area of the triangle is given by A = .5 s(squared) sin (theta). (Done)
b) if theta is increasing at the rate of .5 radian per minute, find the rates of change of the area when (theta)= pi/6 and (theta)/3
c) explain why the rate of change of the area of the triangle is not constant even though d(theta)/dt is constant.

Answer 1

This is a related rates problem. We're given the rate of change for one variable and need to find how that translates into a rate of change for another variable. Specifically, we're given d(theta)/dt and asked to find dA/dt. We solve by finding another rate of change that bridges the gap:\[\frac{ d \theta }{ dt }\frac{ d? }{ d? }=\frac{ dA }{ dt }\]If we treat these as fractions, it's fairly easy to see what rate of change will make this work:\[\frac{ d \theta }{ dt }\frac{ dA }{ d\theta }=\frac{ dA }{ dt }\]So we need to find dA/d(theta) and multiply by d(theta)/dt (which is given as .5) to arrive at dA/dt.

Fortunately we already have an equation for A in terms of theta, so the next step is to find the derivative of that formula. This is fairly easy, using the product rule:\[\frac{ dA }{ d \theta }=\frac{ 1 }{ 2 }(s^2\cos \theta+2s \sin \theta)\]That's a little messy, but there isn't much we can do about that. There's no way to simplify with a trig identity because we have s^2 for one coefficient and 2s for the other.

At this point it's easy to make the mistake of evaluating this expression and thinking we've found the answer. We have to remember to multiply this expression, which is dA/d(theta), times d(theta)/dt, which is .5. The rate of change we're looking for is the expression above times 1/2.

Now you simply have to plug the values of sine and cosine for the relevant angles into this formula and evaluate the result, a task I'll leave to you.

There are various ways to answer the last question, but the simplest is to say that the rate of change in the area is a constant (d(theta)/dt) times something that is not a constant (dA/d(theta)) so it is not constant.