CBSE Class 12 Mathematics 2023: question 32(a)

Hints

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Step-by-step solution

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1. In an equilateral triangle ABC, the median from A is also the altitude. Let $h$ be the length of the median and $x$ the side.

   (Figure: equilateral triangle ABC with the median AD drawn to the midpoint D of BC, perpendicular to BC.)

   1 mark

2. In right $△ADB$: $h^2=x^2-{\left(\frac{x}{2}\right)}^2=\frac{3x^2}{4}$.

   So $h=\frac{\sqrt{3}}{2}x$, i.e. $x=\frac{2h}{\sqrt{3}}$ ...(i)

   2 marks

3. Given $\frac{dh}{dt}=2\sqrt{3}$ cm/s. Differentiating (i) with respect to $t$:

   $\frac{dx}{dt}=\frac{2}{\sqrt{3}}\frac{dh}{dt}$

   1 mark

4. $\frac{dx}{dt}=\frac{2}{\sqrt{3}}\times 2\sqrt{3}=4$

   The side is increasing at the rate of 4 cm/s.

   1 mark

Answer: 4 cm/s

Revise the chapter

• Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)