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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 2cm/ sec. The rate at which the area increases, when side is 10cm 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. (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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