Question

In a Mahakumbh, a drone camera is moving along $3y = x^3 - 3$. When y-coordinate changes 9 times as fast as x-coordinate, it captures good quality pictures. Then one of the precise positions of the drone at that instant is

• A. $(-3, 8)$
• B. $(3, -8)$
• C. $(3, 8)$
• D. $(-3, -8)$

Hint:

"Rate of change" problems almost always involve differentiating an equation with respect to time ($t$). Translating the English phrase "$A$ changes $k$ times as fast as $B$" directly into the equation $\frac{dA}{dt} = k \cdot \frac{dB}{dt}$ is the crucial first step.

Answer 1

The Correct Option is C

1. The equation of the path followed by the drone camera is given by \(3y = x^3 - 3\). For simpler manipulation, we rearrange it to \(y = \frac{x^3 - 3}{3}\).
2. It's stated that the y-coordinate changes 9 times as fast as the x-coordinate, meaning \(\frac{dy}{dt} = 9 \frac{dx}{dt}\).
3. To find \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) in terms of \(x\), we differentiate the equation of the path with respect to \(t\):
   • Given \(y = \frac{x^3 - 3}{3}\), differentiate both sides with respect to \(t\):
     • \(\frac{dy}{dt} = \frac{1}{3} \cdot \frac{d}{dt}(x^3 - 3)\)
     • \(\frac{dy}{dt} = \frac{1}{3} \cdot 3x^2 \frac{dx}{dt} = x^2 \frac{dx}{dt}\)
4. According to the condition \(\frac{dy}{dt} = 9 \frac{dx}{dt}\), substitute the expression obtained from the derivative:
   • \(x^2 \frac{dx}{dt} = 9 \frac{dx}{dt}\)
   • If \(\frac{dx}{dt} \neq 0\), then \(x^2 = 9\).
   • Solving \(x^2 = 9\) gives \(x = \pm 3\).
5. Substitute these \(x\) values back into the path equation to find the corresponding \(y\) values:
   • When \(x = 3\):
     • \(y = \frac{3^3 - 3}{3} = \frac{27 - 3}{3} = \frac{24}{3} = 8\)
     • This gives the point \((3, 8)\).
   • When \(x = -3\):
     • \(y = \frac{(-3)^3 - 3}{3} = \frac{-27 - 3}{3} = \frac{-30}{3} = -10\)
     • This calculation, however, results in \((-3, -10)\)
6. Thus, among the given options, the precise position of the drone that satisfies both the movement path and speed condition is \((3, 8)\).

Therefore, the correct answer is Option (3, 8).