MCQ
The adjacent sides of a rectangle with given perimeter as $100\, cm $ and enclosing maximum area are
- A$10 \,cm$ and $40\, cm$
- B$20 \,cm$ and $ 30\, cm$
- ✓$25\, cm $ and $25 \,cm$
- D$15 \,cm$ and $ 35 \,cm$
✓
Answer
Correct option: C.
$25\, cm $ and $25 \,cm$
c
(c) $2x + 2y = 100 \Rightarrow x + y = 50$…..$(i)$
(c) $2x + 2y = 100 \Rightarrow x + y = 50$…..$(i)$
Let area of rectangle is $A;$ $\therefore A = xy \Rightarrow y = \frac{A}{x}$
Put in $(i),$ we have $x + \frac{A}{x} = 50 \Rightarrow A = 50x - {x^2}$
==> $\frac{{dA}}{{dx}} = 50 - 2x$
For maximum area $\frac{{dA}}{{dx}} = 0$
$\therefore 50 - 2x = 0 \Rightarrow x = 25$ and $y = 25$
Hence adjacent sides are $25$ and $ 25\, cm.$
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