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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

The biomass B(t) of a fish population is the total mass of the members of the population at time t. It is the product of the number of individuals N(t) in the population and the average mass M(t) of a fish at time t. In the case of guppies, breeding occurs continually. Suppose that at time t = 4 weeks the population is 820 guppies and is growing at a rate of 50 guppies per week, while the average mass is 1.2 g and is increasing at a rate of 0.14 g/week. At what rate is the biomass increasing when t = 4?

To determine

To find: The rate at which the biomass is increasing when t=4 week.

Explanation

Given:

The biomass B(t) of fish population is the product of the number of individuals N(t) in the population at time t and the average mass M(t) of a fish at a time t.

That is, B(t)=N(t)M(t).

The population at t=2 weeks is 820 guppies. That is, N(4)=820.

The population is growing at a rate of 50 guppies per week. That is, N(t)=50.

The average mass is 1.2 gram. That is, M(4)=1.2.

The mass is increasing at a rate of 0.14 gram per week. That is, M(t)=0.14.

Derivative rule:

Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

Calculation:

Obtain the rate at which the biomass is increasing when t=4.

The derivative of B(t)=N(t)M(t) is B(t), which is obtained as follows,

B(t)=ddt(N(t)M(t))

Apply the Product rule (1) and simplify the terms,

B(t)=N(t)

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