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Question

Mathematics Question on Application of derivatives

The weight WW of a certain stock of fish is given by W=nw,W= nw, where n is the size of stock and w is the average weight of a fish. If nn and ww change with time tt as n=2t2+3n=2t^{2}+3 and w=t2t+2,w=t^{2}-t+2, then the rate of change of WW with respect to tt at t=1t = 1 is

A

11

B

88

C

1313

D

55

Answer

1313

Explanation

Solution

Let W=nwW = nw dWdt=ndwdt+w.dndt...(1)\Rightarrow \frac{dW}{dt}=n \frac{dw}{dt}+w. \frac{dn}{dt}\,...\left(1\right) Given : w=t2t+2w = t^{2} - t + 2 and n=2t2+3n = 2t^{2}+ 3 dwdt=2t1anddndt=4t\Rightarrow \frac{dw}{dt}=2t-1 and \frac{dn}{dt}=4t \therefore Equation (1)\left(1\right) dwdt=(2t2+3)(2t1)+(t2t+2)(4t)\Rightarrow \frac{dw}{dt} =\left(2t^{2}+3\right)\left(2t-1\right)+\left(t^{2}-t+2\right)\left(4t\right) Thus, dWdtt=1=(2+3)(21)+(2)(4)\frac{dW}{dt}|_{t=1}=(2+3)(2-1)+(2)(4) =5(1)+8=13= 5 (1) + 8 = 13