Dead leaves accumulate on the ground in a forest at a rate of 4 grams per square centimeter per year. At the same time, these leaves decompose at a continuous rate of 70 percent per year. A. Write a differential equation for the total quantity Q of dead leaves (per square centimeter) at time t: dQ dt 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? Q(0) What is the equilibrium level? Q Does the equilibrium value attained depend on the initial condition? ○ A. yes B. no

The von Bertalanffy growth model states that the rate of increase in the length L(t) of a shark of age t in years is proportional to the difference between a fixed maximum length M and its current length L(t). 1. Write a differential equation (DE) for the length of the shark at any time, using to represent the proprotional constant. (Hint: your answer should be of the form L'(t) = ....) 2. Explain carefully what this DE predicts would happen if a mutant shark were born with a length larger than M. 3. Show that L(t) = M – (M — Lo)e¯rt is a solution to the DE in Part 1, where Lo is the length of the shark at time t = 0 when it is born. (Hint: if y(t) = A + Be¯rt then y' (t) = −rBe¯rt.)

Please help for c and d.

2. (10%) Consider a tank filled with water with a draining tap at the bottom. The water level in the tank is a function of the time, h(t). It has been observed that the rate of change of h with respect to time is proportional to the square root of the water level h at a given time. Write the differential equation for the water level h(t) at a given time t. h(t) Тар exit