Question:
The weight $W$ of a certain stock of fish is given by $W= nw,$ 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 $n=2t^{2}+3$ and $w=t^{2}-t+2,$ then the rate of change of $W$ with respect to $t$ at $t = 1$ is
The weight $W$ of a certain stock of fish is given by $W= nw,$ 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 $n=2t^{2}+3$ and $w=t^{2}-t+2,$ then the rate of change of $W$ with respect to $t$ at $t = 1$ is
Updated On: Aug 15, 2022
- $1$
- $8$
- $13$
- $5$
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The Correct Option is C
Solution and Explanation
Let $W = nw$
$\Rightarrow \frac{dW}{dt}=n \frac{dw}{dt}+w. \frac{dn}{dt}\,...\left(1\right)$
Given : $w = t^{2} - t + 2$ and $n = 2t^{2}+ 3$
$\Rightarrow \frac{dw}{dt}=2t-1 and \frac{dn}{dt}=4t$
$\therefore$ Equation $\left(1\right)$
$\Rightarrow \frac{dw}{dt} =\left(2t^{2}+3\right)\left(2t-1\right)+\left(t^{2}-t+2\right)\left(4t\right)$
Thus, $\frac{dW}{dt}|_{t=1}=(2+3)(2-1)+(2)(4)$
$= 5 (1) + 8 = 13$
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
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- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
Read More: Application of Derivatives