The population of a variety of truffle increases at an annual rate euqal to 3% o

Question

The population of a variety of truffle increases at an annual rate euqal to 3% of the current population P. Measured in kilograms per acre: this takes into account the birth rate and death rate due to natural casues. Meanwhile humans harvest 0.5 kg/acre each year.

A. Write a differential equation that describes the rate of change of the truffle population with respect to time. Taking both the natural increase and the lose to human consumption into account.

B. If the current population is 10 kg/acre, use Euler's method with two steps to approximate the population four years from now.

C. Scarcity of the truffles leads to a call for halting the harvesting unit the population readies 12 Kg/acre. What amount should then be harvested each year to exactly maintain this population?

D. Assume that the population is 10kg/acre and there is no harvesting. So the truffles increase simply at a rate equal to 3% of the current population. Give a formula for P as a function of time. Then determine how long it will be unitl the population reaches 12 kg/arce

Show work please

Explanation / Answer

a) you should have gotten P = (80/3) e^(0.03t) - 50 / 3 , assuming P(0) = 10 {from b) }

where dP / dt = incoming - outgoing

c) you desire P = 12..and 12 kg/acre is achieved when t ? 2.4 years ;

thus harvest until dP / dt = 0where dP/dt = 0 when P = 12 { incoming = outgoing }

0.03 ( 12 ) = 0.36...thus harvest 0.36 kg/acre/yr after year 2.4 .

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