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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
MCQ
Choose the correct answer from the given four options :
The sides of an equilateral triangle are increasing at the rate of $2\ cm/ \sec$. The rate at which the area increases, when side is $10\ cm$ is :
  • A
    $10\text{ cm}^2\text{/s}$
  • B
    $\sqrt{3}\text{ cm}^2\text{/s}$
  • ✓
    $10\sqrt{3}\text{ cm}^2\text{/s}$
  • D
    $\frac{10}{3}\text{ cm}^2\text{/s}$
✓

Answer

Correct option: C.
$10\sqrt{3}\text{ cm}^2\text{/s}$
Let the side of an equilateral triangle be $x \ cm,$
$\therefore$ Area of equilateral triangle, $\text{A}=\frac{\sqrt{3}}{4}\text{x}^2\ \ \dots(\text{i})$
Also, $\frac{\text{dx}}{\text{dt}}=2\text{ cm/s}$
On differentiating Eq. $\text{(i) w.r.t. t},$ we get
$\frac{\text{dA}}{\text{dt}}=\frac{\sqrt{3}}{2}\cdot2\text{x}\cdot\frac{\text{dx}}{\text{dt}}$
$=\frac{\sqrt{3}}{4}\cdot2\cdot10\cdot2$
$\Big[\because\ \text{x}=10\text{ and }\frac{\text{dx}}{\text{dt}}=2\Big]$
$=10\sqrt{3}\text{ cm}^2\text{/s}$

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