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Dead leaves accumulate on the ground in a forest at a rate of 3 grams per square centimeter per year. At the same time, these leaves decompose at a continuous rate of 75 percent per year.

A. Write a differential equation for the total quantity, Q, of dead leaves (per square centimeter) at time t.

B. Sketch a solution to your differential equation showing that the quantity of dead leaves tends toward an equilibrium level. Assume that initially (t=0) there are no leaves on the ground.

What is the initial quantity of leaves?

What is the equilibrium level?

Does the equilibrium value attained depend on the initial condition?

A. yes
B. no

Respuesta :

Answer:

A - [tex]\frac{dD}{dt}[/tex] = 3 - 0.75D

B-  4 grams per square centimeter.

C-  Yes

Step-by-step explanation:

  • D is dead leaves at time t. from the given statement we can write differential equation as:

        [tex]\frac{dD}{dt}[/tex] = 3 - 0.75D

  • Need to solve this differential equation. Solved equation attached.

        if we put t = 0 in the equation we will get C = -4. This will give us an       equation:

          D = 4(1 - [tex]e^{-0.75t}[/tex])

  • From the graph attached you can see that equilibrium quantity is 4 grams per square centimeter.
  • Yes it does since a different initial condition will change the value of C and hence the value of equilibrium.

Ver imagen saadahmed81994
Ver imagen saadahmed81994

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