Chapter 1 Exercise - wavelength, accelerationgdue to gravity and de...

Chapter 1 (Exercise) 1) The velocityuof propagation of deep ocean waves is a function of the wavelength, accelerationgdue to gravity and densityof the liquid.By assuming thatu=k a g b

c , wherekis a dimensionless constant,a; b; c are real numbers, nd the relation betweenu; ; gand.2) Which of the following equations contain an error and which are dimen- sionally correct?(i)v 2 =u 2

• 2gz,

(ii)p= lu z , (iii)F=pA, (iv)p+ 1 2 u 2 =mgz, wherepis the pressure,is the density,uandvare velocities,gis the acceleration due to gravity,Fis a force,mis a mass,Ais an area,zand lare lengths.3) A man plans to build a garden, triangular in shape, such that the height of the triangle is 4 ft shorter than the length of the base and also plans to put a fence along the hypotenuse of the triangle. Write an equation to model the length of the hypotenuse and obtain the value (in ft) that will minimize the length of the hypotenuse.4) The time of oscillationT1of a small drop of liquid under surface tension depends on the density, the radiusrand the surface tensionsand is given byT1= s r 3 s . Check weather the equation is dimensionally correct or not.5) Suppose a company decides to make plastic container, which can hold

0:001m

3 (1 litre) of some powder. The container should have a circular base, with double thickness at the top and the bottom. Let the cost of the

plastic sheet be $0:20 per square metre. Using the concept of mathemati-

cal modeling, obtain the most economical size of the plastic container for the powder.6) A person running a business, makes an initial investment of $2000 and the cost of manufacturing each item is $15. Write a function to model the total costs. Also, write a function to model the average cost per item.Graphically, nd the minimum of this function and interpret the result.7) Due to the increase in global population, many environmentalists be- lieve that the carrying capacity of the Earth (the maximum population 2 Solutions Manual for Mathematical Modeling Models, Analysis and Applications, 2e by Sandip Banerjee (All Chapters) 1 / 4

Earth can sustain) depends not only on the nite natural resources of the planet-water, land, air, and materials, but also on how the people use and preserve the resources. The graphs show four dierent ways that a growing population can approach its carrying capacity over time. Explain each of the scenarios by analyzing the graphs.8) In an environment, a new species has been introduced, which can aect the growth of an existing species in various ways. The following graphs show four hypothetical scenarios after Species A is introduced into an environment where Species B is already present and living. Explain each of the scenarios by analyzing the graphs.9) Obtain the solution of the Gompertz growth model by solving the dier- ential equation dx dt =rxln

k x !; x(0) =x0; analytically. Hence, obtain the limiting value ofx(t) ast! 1.10) Four dierent functions are described in the gures A, B, C and D. Match each description with the appropriate graph.(i) The concentration of the pollutant that pours into a lake is a function of time and initially increases quite rapidly, but due to the natural mix- ing and self-cleansing action of the lake, the concentration levels o and stabilizes at some saturation level.(ii) A train travels at a constant speed and its distance from the point of origin is a function of time.(iii) The population of a city, which is a function of time, starts increasing rather slowly, but continues to grow at a faster and faster rate.(iv) The production level at a manufacturing plant is a function of the amount of money invested in the plant (capital outlay). At rst, small increases in capital outlay result in large increases in production, but eventually the investors begin to experience diminishing returns on their money, so

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that although production continues to increase, it is at a disappointingly slow rate.11) The reaction of lead nitrate and potassium iodide in solution is lead iodide, which precipitates as a yellow solid at the bottom of the container.More lead iodine is produced if you add more lead nitrate to the solution and continues till all the potassium iodide is used up. The table shows the height of the precipitate in the container as a function of the amount of lead nitrate added [17].Lead nitrate solution (cc)0.51.01.52.02.53.03.54.0 Height of precipitate (mm)2.84.86.27.49.59.69.69.6 (i) Plot the data and sketch a piecewise linear function with two parts to t the data points.(ii) Write a mathematical formula for the piecewise function.(iii) Interpret your graph in the context of the problem.12) A person goes into skydiving and let f(t) approximates his altitude as a function time t (in seconds), starting from ground, till he reaches the ground.f(t) = 30t;0t <300; = 9000;300t <400; = 900016(t400) 2 ;400t <420;

= 2600130(t420);420t440:

(i) Draw the graph of f(t) and explain, what is happening during each inter- val.(ii) There are two times that the person is at an altitude of 3600 ft, nd them.13) A person runs a business of wooden jewelery boxes. The cost of manufac- turing the wooden box is $6 per box, plus an initial investment of $250 is needed for other materials. Another $3 is needed to decorate the boxes.Write a function to model the total cost of the boxes.14) Consider the damped Lokta Volterra predator prey model dx dt =rx(1 x k )xy; dy dt =y+xy; x(0) =x0; y(0) =y0;

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wherexandyare densities of prey and predator respectively,ris the intrinsic growth rate of the prey andkis its carrying capacity (maximum resources the environment can sustain indenitely),is the predation rate per individual per unit time,is the reproduction rate of predators per individual per unit time. Here,r; k; ; ; are positive parameters and the initial conditions arex(0) =x0,y(0) =y0. Non-dimensionalize the model.15) A chocolate making company markets its products to individuals, to re- tailers, and to wholesale distributors. Three dierent prices are oered by the company for its chocolates. If you order 15 or fewer boxes, the price

is $4:50 each. If you order more than 15 but no more than 60 boxes, the

price is $3:75 each. If you order more than 60 boxes, the price is $2:75 each. Write a functionf(x) to model the orderingxnumber of boxes of chocolates and plot the graph off(x).

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