Question

The coordinates of the points on the curve \(4y = x^2\) that are nearest to the point \((0,5)\) are ...

• A. \((-2\sqrt{3},3)\)
• B. \((2\sqrt{3},-3)\)
• C. \((3,2\sqrt{3})\)
• D. \((2\sqrt{3},2)\)

Hint:

Relate dS/dt to dr/dt using S = 4 pi r squared.

Answer 1

The Correct Option is A

Step 1: Setup:
Surface area \(S = 4\pi r^2\). Differentiating with time, \(\frac{dS}{dt} = 8\pi r\frac{dr}{dt}\).

Step 2: Find r:
When \(S = 16\pi\), \(4\pi r^2 = 16\pi\), so \(r = 2\) cm.

Step 3: Find dr/dt:
\[ 4\pi = 8\pi(2)\frac{dr}{dt} \Rightarrow \frac{dr}{dt} = \frac{4\pi}{16\pi} = 0.25\text{ cm/s} \]
Here the area grows at a steady \(4\pi\) cm\(^2\)/s, but as the ball gets bigger, the same increase in area needs a smaller increase in radius. That is why the answer is smaller than 1 cm/s. The values 0.5 and 0.125 would come from using \(r = 1\) or \(r = 4\) by mistake.

Final Answer:
The radius increases at 0.25 cm/s, option (B). \[ \boxed{0.25\text{ cm/s}} \]