Q54 / 90 · JEE Main 2012 (19 May Online) · Mathematics

The weight $W$ of a certain stock of fish is given by $W=n w$, where $n$ is the size of stock and $w$ is the average weight of a fish. If $n$ and $w$ change with time $t$ as $n=2 t^2+3$ and $w=t^2-t+2$, then the rate of change of $W$ with respect to $t$ at $t=1$ is

  • A
    1
  • B
    8
  • Correct answer
    C
    13
  • D
    5

Solution

Let $\mathrm{W}=n w$ $$ \Rightarrow \frac{d W}{d t}=n \frac{d w}{d t}+w \cdot \frac{d n}{d t} $$ Given : $w=t^2-t+2$ and $n=2 t^2+3$ $\Rightarrow \frac{d w}{d t}=2 t-1$ and $\frac{d n}{d t}=4 t$ $\therefore$ Equation (1) $$ \Rightarrow \frac{d w}{d t}=\left(2 t^2+3\right)(2 t-1)+\left(t^2-t+2\right)(4 t) $$ Thus, $\left.\frac{d W}{d t}\right|_{t=1}=(2+3)(2-1)+(2)(4)$ $$ =5(1)+8=13 $$

Hence, the correct answer is option: 'C'.

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