Multiple choice

If $K$ is a diameter of a circle and $A$ is the area of sector of the circle whose vertical angle is $\theta$ then $\frac {dA}{dt}=$

  1. $\dfrac{{{k^2}}}{8}\left( {\dfrac{{d\theta }}{{dt}}} \right)$
  2. $\left( {\dfrac{{{k^2}}}{4}} \right)\left( {\dfrac{{d\theta }}{{dt}}} \right)$
  3. $\dfrac{d{\theta}}{dt}$
  4. $k\left( {\dfrac{{d\theta }}{{dt}}} \right)$
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A Correct answer
Explanation

Area of sector A = (1/2) * r^2 * theta. Since diameter K = 2r, r = K/2. A = (1/2) * (K/2)^2 * theta = (K^2 / 8) * theta. Differentiating with respect to t: dA/dt = (K^2 / 8) * (d(theta)/dt).

AI explanation

The radius of the circle is r = K / 2, where K is the diameter. The formula for the area of a sector with vertical angle theta is A = (1/2)r^2 theta. Substituting the radius gives A = (1/2)(K / 2)^2 theta, which simplifies to A = (K^2 / 8) theta. Differentiating both sides with respect to time t yields dA/dt = (K^2 / 8)(d theta / dt).

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