Question

How do you do this question?

Answer 1

Answer:

23,333 kg

Step-by-step explanation:

G(t) = 30,000 e^(-0.6 t) / (1 + 5 e^(-0.6 t))²

The change in the biomass between 2000 and 2020 is:

∫₀²⁰ G(t) dt

∫₀²⁰ 30,000 e^(-0.6 t) / (1 + 5 e^(-0.6 t))² dt

If u = 1 + 5 e^(-0.6 t), then:

du = -3 e^(-0.6 t) dt

-10,000 du = 30,000 e^(-0.6 t) dt

∫ (-10,000 / u²) du

-10,000 ∫ u⁻² du

10,000 u⁻¹ + C

10,000 / (1 + 5 e^(-0.6 t)) + C

Evaluate between t=0 and t=20.

10,000 / (1 + 5 e^(-0.6 × 20)) − 10,000 / (1 + 5 e^(-0.6 × 0))

10,000 / (1 + 5 e^(-12)) − 10,000 / 6

≈ 8333

The biomass increases by 8,333 kg.  So the final biomass is 23,333 kg.