Let the median of a variable equilateral triangle is increasing at the rate of \( 2\sqrt{3} \) cm/s. Then the rate at which its each side is increasing is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4 cm/s
In an equilateral triangle of side a, the median is also the altitude:
\( \displaystyle m=\sqrt{a^2-\left(\frac{a}{2}\right)^2} \)
\( \displaystyle =\frac{\sqrt{3}}{2}a \)
Differentiating with respect to t,
\( \displaystyle \frac{dm}{dt}=\frac{\sqrt{3}}{2}\cdot\frac{da}{dt} \)
\( \displaystyle 2\sqrt{3}=\frac{\sqrt{3}}{2}\cdot\frac{da}{dt} \)
\( \displaystyle \frac{da}{dt}=4\ \text{cm/s} \)
Why the other options are incorrect
Option (1) \( \sqrt{3} \) cm/s — Incorrect. The side increases at 4 cm/s, not \( \sqrt{3} \) cm/s.
Option (2) \( 2\sqrt{3} \) cm/s — Incorrect. \( 2\sqrt{3} \) cm/s is the rate of the median itself; the side changes faster, since \( m=\frac{\sqrt{3}}{2}a \) is shorter than a.
Option (3) 2 cm/s — Incorrect. 2 cm/s is half the correct rate: \( 2\sqrt{3}\div\sqrt{3}=2 \) forgets the factor \( \frac{1}{2} \) in \( m=\frac{\sqrt{3}}{2}a \).