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

Step 1: Understanding the Concept:
This is an application of derivatives involving rates of change. The phrase "y-coordinate changes 9 times as fast as x-coordinate" translates mathematically to a specific relationship between their derivatives with respect to time ($\frac{dy}{dt}$ and $\frac{dx}{dt}$).

Step 2: Key Formula or Approach:
1. Express the given rate condition mathematically: $\frac{dy}{dt} = 9 \cdot \frac{dx}{dt}$. 2. Differentiate the given curve equation $3y = x^3 - 3$ with respect to time $t$. 3. Substitute the rate condition into the differentiated equation to solve for $x$. 4. Substitute the found $x$ values back into the original curve equation to find the corresponding $y$ coordinates.

Step 3: Detailed Explanation:
The equation of the drone's path is: \[ 3y = x^3 - 3 \] Differentiate both sides with respect to time $t$: \[ \frac{d}{dt}(3y) = \frac{d}{dt}(x^3 - 3) \] \[ 3 \frac{dy}{dt} = 3x^2 \frac{dx}{dt} \] Divide both sides by 3: \[ \frac{dy}{dt} = x^2 \frac{dx}{dt} \] We are given the condition that the y-coordinate changes 9 times as fast as the x-coordinate. This means: \[ \frac{dy}{dt} = 9 \frac{dx}{dt} \] Substitute this condition into our differentiated equation: \[ 9 \frac{dx}{dt} = x^2 \frac{dx}{dt} \] Assuming the drone is actually moving, $\frac{dx}{dt} \neq 0$, so we can divide both sides by $\frac{dx}{dt}$: \[ 9 = x^2 \] This gives two possible values for the x-coordinate: \[ x = \pm 3 \] Now, find the corresponding y-coordinates by substituting these x values back into the original path equation $3y = x^3 - 3$. Case 1: If $x = 3$: \[ 3y = (3)^3 - 3 \] \[ 3y = 27 - 3 \] \[ 3y = 24 \] \[ y = 8 \] So, one position is $(3, 8)$. Case 2: If $x = -3$: \[ 3y = (-3)^3 - 3 \] \[ 3y = -27 - 3 \] \[ 3y = -30 \] \[ y = -10 \] So, another position is $(-3, -10)$. Looking at the given options, $(-3, -10)$ is not present, but $(3, 8)$ is.

Step 4: Final Answer:
One of the precise positions is $(3, 8)$.