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 =

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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
A certain radioactive substance has a half-life of 3.6 years. Find how long it takes (in years) for the substance to decompose to 10% of its initial amount.
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.
Question 16 Suppose that you have downloaded data on the daily share prices for a company for the period from the start of 2020 to the end of 2020 from the website of Yahoo Finance UK. You have also calculated the daily rates of return for the company's shares as (closing price - opening price)/opening price. By using the knowledge that you have gained from descriptive statistics, how can you obtain useful information about the company's share performance in terms of risk and return using a range of alternative indicators? Note that you should give necessary formulas for the technical concepts wherever they may apply.
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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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