S= 6- \ Dead leaves accumulateon the ground at a rateof 4 grams per squarecentimeter per year, theydecompose at a continuousrate of 65% per year-a)write a differential equation1 for the total quantity ofdead leaves at time t42b) what is the initial quantityof leaves? Q(0) =Lc) what is the equilibriumlevel? Qeq =

Answered: Image /qna-images/answer/27cc674b-3159-4560-84f2-46b08756551b.jpg

Answered: Dead leaves accumulate on the ground at…

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.)
The population of country doubles in fifty years assuming that the rate of increase is proportional to the number of inhabitants state this information in a differential equation if x denotes the population at time t. Hence solve
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.
Show that solving the logistic differential equation results in the logistic growth function
Use mathematical induction to prove the following statement n ≥ 1 ∑ n i=1 i3 = n2 (n + 1) 2/ 4
At t=0, a 600 liter tank initially contains 50 grams of salt dissolved in 300 liters of water. Then, water that contains 2 grams of salt per liter is poured into the tank at a rate of 5 liters per minute. The solution in the tank is constantly stirred so the salt concentration is uniform at any given time t. Water leaves the tank at a rate of 2 liters per minute. Determine when the tank will overflow and create a differential equation to determine how much salt will be in the tank after 1 hour.
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

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