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STD 12 - 6. Application of derivatives — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 6. Application of derivatives1 Mark
MCQ
A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is
  • A
    $\pi {x^2}$
  • B
    $\frac{3}{2}{x^2}$
  • C
    $\sqrt {\frac{{{x^3}}}{8}} $
  • ✓
    $\frac{1}{2}{x^2}$
✓

Answer

Correct option: D.
$\frac{1}{2}{x^2}$
d
(c) $: A T=x \sin \alpha$

$B T=x \cos \alpha$

Area of triangle

$\mathrm{ABC}=\frac{1}{2}$ base $\times$ height

$=\frac{1}{2}(2 B T)(A T)$

$=\frac{1}{2}\left(2 x^{2} \cos \alpha \sin \alpha\right)$

$=\frac{1}{2} x^{2} \sin 2 \alpha \leq \frac{1}{2} x^{2}$ as $-1 \leq \sin 2 \alpha \leq 1$

Maximum are of $\Delta A B C=\frac{1}{2} x^{2}$

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