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:dQdtB. 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? QDoes the equilibrium value attained depend on the initial condition?○ A. yesB. no

Part A: Write a differential equation for the total quantity Q of dead leaves (per square…

Answered: Dead leaves accumulate on the ground in…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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.)
Population of the world, (x) is assumed to be a continuous variable at time t years. The rate of increase of x due to birth is ax and the rate of decrease of x due to death is Bx, where a and B are positive constant. a. By applying differential equation concept and assuming a = 0.060 and B = 0.040. Show the population of the world (x) double itself in approximately 35 years. Justify your answer. b. By using appropriate scale, draw the graph of world population growth.
Waste material from a mining operation is dumped on a “slag-heap”, which is a large mound, roughly conical in shape, which continually increases in size as more waste material is added to the top. In a mathematical model, the rate at which the height h of the slag-heap increases is inversely proportional to h2.I. Express this statement as a differential equation relating h with time t. II. Show by integration that the general solution of the differential equation relating h and t may be expressed in the form h3 = At + B, where A and B are constants. III. A new slag- heap was started at time t = 0, and after 2 years its height was 18m. Find the time by which its height would grow to 30m.
Please help for c and d.
A 100,000 liter tank of pure water was accidentally mixed with a 7000 liters of salt with a concentration of 90k kilos. To solve this problem, they continuously pour fresh water into the tank at a rate of 50 liters per minute and release the salt solution at the same rate. Find: a.The differential equation on this problem. b.The amount of salt after 3 days. c.Number of minutes at half of the original salt content.
A tank originally contains 120 litre of brine with a salt concentration of 4 gram of salt per litre. Then brine with 15 gram of salt per litre is poured into the tank at a rate of 6 litre/minute, and the well-stirred mixture is allowed to leave at the same rate. a. Model this situation by a differential equation with initial conditions. b. Find the concentration of salt in the tank after 20 minutes. c. Find the eventual concentration of salt in the tank. (Hint: The eventual value will be as time goes to infinity.)
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
A tank initially contains 200L of pure water. Salt enters into the solution constantly 5 kilograms for every minute, then will be mixed well into the contents of the tank. If the well-mixed solution exits from the tank at the rate 10 L per minute, show how the differential equation below describes the salt concentration S in the solution at any time t. ds :5 t - 20 %3! dt

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