Multiple choice

A variable triangle $ABC$ is inscribed in a circle of diameter $x$ units. If at a particular instant the rate of change of side, '$a$' is $x/2$ times the rate of change of the opposite angle $A$, then $A$ =

  1. $ \pi /6 $
  2. $ \pi /3 $
  3. $ \pi /4 $
  4. $ \pi /2 $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the Law of Sines, a = x * sin(A). Differentiating with respect to time, da/dt = x * cos(A) * dA/dt. Given da/dt = (x/2) * dA/dt, we have x * cos(A) = x/2, so cos(A) = 1/2, which means A = pi/3.

AI explanation

Using the extended sine rule, side a equals x multiplied by sin A, where x is the diameter of the circumscribed circle. Differentiating both sides with respect to time gives the rate of change of side a as x multiplied by cos A, which is then multiplied by the rate of change of angle A. Setting this equal to x/2 times the rate of change of angle A results in the equation x multiplied by cos A equals x/2. Solving for A gives cos A = 1/2, so A equals pi/3. The correct answer is pi/3.

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