#### To determine

**To find:** the biomass for the year 2020, given G(t)=60000e−0.6t(1+5e−0.6t)2

#### Answer

biomass for the year 2020 = 41,666 kg

#### Explanation

Given that the biomass is 25,000 kg in the year 2000.

The growth rate of a fish population was modelled by the equation G(t)=60000e−0.6t(1+5e−0.6t)2 where t is measured in years and G in kilograms per year.1 The units on the growth rate function are kg per year and the units on t (and then ∆t and dt) are years, so the integral will give the net change in kg per year /years = kg from 2000 to 2020.

The net change from 2000 to 2020 = ∫02060000e−0.6t(1+5e−0.6t)2dt

Let u=1+5e−0.6t,then du=5e−0.6t(−0.6)dt=−3e−0.6tdt

When t=0;u=6 and t=20;u=1+5e−0.6(20)=1+5e−12

So we get,

∫02060000e−0.6t(1+5e−0.6t)2dt=∫61+5e−1260000.u−15u2.du−3.u−15=−20000∫61+5e−12duu2=−20000[−1u]61+5e−12=20000[11+5e−12−16]=20000×0.8333≈16666

The change in the biomass from 2000 to 2020 is 16,666 kg. Thus, the predicted biomass for 2020 is about 25,000 + 16,666 = 41,666 kg.

**Conclusion:** biomass for the year 2020 = 41,666 kg