Consider a rectangle whose length is increasing at the uniform ra… - Vidyadip

MCQ

Consider a rectangle whose length is increasing at the uniform rate of $2\, m/sec$, breadth is decreasing at the uniform rate of $3\, m/sec$ and the area is decreasing at the uniform rate of $5\,m^2/ sec$ . If after some time the breadth of the rectangle is $2\, m$ then the length of the rectangle is ........ $m.$

A
$2$
B
$4$
C
$1$
✓
$3$

✓

Answer

Correct option: D.

$3$

d
Let $A$ be the area, $b$ be the breadth and $\ell $ be the length of the rectangle.

Given: $\frac{{dA}}{{dt}} = - 5,\frac{{d\ell }}{{dt}} = 2,\frac{{db}}{{dt}} = - 3$

We know, $A = \ell \times b$

$ \Rightarrow \frac{{dA}}{{dt}} = \ell .\frac{{db}}{{dt}} + b.\frac{{d\ell }}{{dt}} = - 3\ell + 2b$

$ \Rightarrow - 5 = 3\ell + 2b$

When $b=2$, we have

$ - 5 = 3\ell + 4 \Rightarrow \ell = \frac{9}{3} = 3m$

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