Source: Chapra 5

th edition

I) MATHEMATICAL MODELING AND

ENGINEERING PROBLEM SOLVING

1. Figure depicts the various ways in which

an average man gains and loses water in

one day. One liter in ingested as food,

and the body metabolically produces

0.3L. In breathing air, the exchange is

0.05L while inhaling, and 0.4L while

exhaling over a one-day period. The

body will also lose 0.2, 1.4, 0.2, and

0.35L through sweat, urine, feces, and

through the skin, respectively. In order

to maintain steady-state condition, how

much water must be drunk per day?

2. For free-falling parachutist with linear

drag, assume a first jumper is 70kg and

has a drag coefficient of 12kg/s. If a

second jumper has a drag coefficient of

15kg/s and a mass of 75kg, how long will

it take him to reach the same velocity

the first jumper reached in 10s?

3. The amount of a uniformly distributed

radioactive contaminant contained in a

closed reactor is measured by its

concentration c (Becquerel/liter of Bq/L).

The contaminant decreases at a decay

rate proportional to its concentration-

that is

Decay rate = -kc

where k is a constant with units of day-1

.

Therefore, a mass balance for the

reactor can be written as

a. Use Euler’s method to solve this

equation from t=0 to 1d with

k=0.2d-1

. Employ a step size of

Δt=0.1. The concentration at t=0 is

10Bq/L.

b. Plot the solution on a semilog graph

(i.e., ln c versus t) and determine the

slope. Interpret your results.

4. Newton’s law of cooling says that the

temperature of a body changes at a rate

proportional to the difference between

its temperature and that of the

surrounding medium (the ambient

temperature),

Where T= the temperature of the body

(°C), t= time (min), k= the proportionality

constant (per minute), and = the

ambient temperature (°C). Suppose that

a cup of coffee originally has a

temperature of 68°C. Use Euler’s

method to compute the temperature

from t = 0 to 10 min using a step size of

1min if =21°C and k=0.017/min.

5. Suppose that a spherical droplet of liquid

evaporates at a rate that is proportional

to its surface area

Where the volume (mm3), t = time (h),

k= the evaporation rate (mm/hr), and A=

Metabolism

Urine

Skin

Feces

Air

Sweat Drink

Food

Source: Chapra 5

th edition

surface area (mm

2). Use Euler’s method

to compute the volume of the droplet

from t = 0 to 10min using step size of

0.25min. Assume that k = 0.1mm/min

and that the droplet initially has a radius

of 3mm. Assess the validity of your

results by determining the radius of your

final computed volume and verifying

that is consistent with the evaporation

rate.

6. A storage tank contains a liquid at depth

y where y=0 when the tank is half full.

Liquid is withdrawn at a constant flow

rate Q to meet demands. The contents

are resupplied at a sinusoidal rate

3Qsin2(t).

Or, since the surface area A is constant

Use Euler’s method to solve for the depth y from

t = 0 to 10d with a step size of 0.5d. The

parameter values are A = 1200 m2 and Q = 500

m3/d. Assume that the initial condition is y = 0.

Approximations and Round-off errors

1. Convert the following base-2 numbers to

base-10: (a) 101101, (b) 101.101, and (c)

0.01101.

2. Evaluate e-5

using two approaches

And compare with the true value of

6.737947x10-3

. Use 20 terms to evaluate

each and compute true and approximate

relative errors as terms are added.

3. (a) Evaluate the polynomial

Y=x3-7x

2+8x-0.35

At x=1.37. Use 3-digit arithmetic with

chopping. Evaluate the percent relative

error.

(b) Repeat (a) but express y as

y=((x-7)x +8)x - 0.35

Evaluate the error and compare with

part (a)

4. Determine the number of terms

necessary to approximate cos x to 8

significant figures using the Maclaurin

series approximation

Calculate the approximation using a

value of x = 0.3π. Write a program to

determine your result.

5. How can the machine epsilon be

employed to formulate a stopping

criterion εs for your programs? Provide

an example.

6. The infinite series

y

0

Source: Chapra 5

th edition

Converge on a value of f(n) = π4/90 as n

approaches infinity. Write a program in

single precision to calculate f(n) for n=

10 000 by computing the sum from i=1

to 10 000. Then repeat the calculation

but in reverse order-that is, from

i = 10 000 to 1 using increments of -1. In

each case, compute the true percent

relative error. Explain the results.

Truncation Errors and the Taylor Series

1. Use zero- trhough third-order Taylor

series expansions to predict f(3) for

f(x) = 25x3 – x

2 + 7x – 88

using a base point at x = 1. Compute

the true percent relative error for

each approximation. Discuss the

meaning of the results.

2. Use forward and backward

difference approximations of O(h2)

to estimate the first derivate of the

function examined in before

exercise. Perform the evaluation at

x=2 using steps sizes of h=0.25 and

0.125. Compare your estimates with

the true of the second derivative.

3. Evaluate and interpret the condition

numbers for

(a) F(x) = +1 for x = 1.00001

(b) F(x) = e-x

for x = 10

(c) F(x) = for x = 200

(d) F(x) = e-x

-1/x for x = 0.001

4. A missile leaves the ground with an

initial velocity Vo forming an angle

Φo with the vertical as shown in

figure. The maximum desired

altitude is αR where R is the radius

of the earth. The laws of mechanics

can be used to show that

2

Where ve = the escape velocity of

the missile. It is desired to fire the

missile and reach the design

maximum altitude within an

accuracy of +/- 2%. Determine the

range of values for Φo if ve/vo = 2

and

5. Consider the function f(x) =x3–2x+4

on the interval [-2,2] with h=0.25.

Use the forward, backward, and

centered finite difference

approximations for the first and

second derivates so as to graphically

illustrate which approximations

along with the theoretical, and do

the same for the second derivative

as well.

Φo

vo

R