**Problems**

Youcan find the answers to problems marked with an asterisk at the end of the text. Be sure to label and format properly any plots required’ by the following problems. Label each axis properly. Use a legend, data markers, or different line types as needed. Choose proper axis scaling and tick-mark spacing. Use a title, a grid, or both if they help to interpret the plot.

**Section 5.1**

- Breakeven analysis determines the production volume at which the total production cost is equal to the total revenue. At the breakeven point, there is neither profit nor loss. In general, production costs consist of fixed costs

and variable costs. Fixed costs include salaries of those not directly

involved with production, factory maintenance costs, insurance costs, and so on. Variable costs depend on production volume and include material costs, labor costs, and energy costs. In the following analysis, assume that we produce only what we can sell; thus the production quantity equals the sales. Let the production quantity be Q, in gallons per year. Consider the following costs for a certain chemical product: Fixed cost: $3 million per year.

Variable cost: 2.5 cents per gallon of product. The selling price is 5.5_Cen!?per gallon. Use this data to plot the total cost and the revenue versus Q, and graphically determine the breakeven point. Fully label the plot and mark the breakeven point. Ear what range of Q is production profitable? For what value of Q is the profit a maximum?

2. Consider the following costs for a certain chemical product: Fixed cost: $2.045 million/year. Variable costs: Material cost: 62 cents per gallon of product. Energy cost:’ 24 cents per gallon of product. Labor cost: 16 cents’per gallon of product.

Assume that we produce only what we sell. Let P be the selling price in dollars per gallon. Suppose that the selling price and the sales quantity Q are interrelated as follows: Q = 6 x 106 – 1.1 x 106P. Accordingly, if we raise the price, the product becomes less competitive and sales drop. Use this information to plot the fixed and total variable costs versus Q, and graphically determine the breakeven point(s). Fully label the plot and mark the breakeven points. For what range of Q is the production profitable? For what value of Q is the profit a maximum?

3.* Roots of polynomials appear in many engineering applications, such as electrical circuit design and structural vibrations. Find the real roots of the polynomial equation

in the range -10 x 10 by plotting the polynomial.

4. To compute the forces in structures, engineers sometimes must solve equations similar to the following. Use the f plot function to find all the positive roots of this equation:

5.* Cables are used to suspend bridge decks and other structures. If a heavy . uniform cable hangs suspended from its two endpoints, it takes the shape of a catenary curve whose equation is

where a is the height of the lowest point on the chain above some

horizontal reference line, x is the horizontal coordinate measured to the right from the lowest point, and y is the vertical coordinate measured up from the reference line. Let a = 10m. Plot the catenary curve for – 20 ~ x ~ 30 m. How high is each endpoint?

6. Using estimates of rainfall, evaporation, and water consumption, the town engineer developed the following model of the water volume in the

where V is the water volume in liters, and t is time in days. Plot Vet)

versus t. Use the plot to estimate how many days it will take before the water volume in the reservoir is 50 percent of its initial volume of 109 L.

7. It is known that the following Leibniz series converges to the value 7T /4 as 11 00.

CHAPTER 5 Advanced Plotting and Model Building

Plot the difference between tt /4 and the sum 5(n) versus 11 for 0 :::

11 ::: 200.

8. A certain fishing vessel is initially located in a horizontal plane at x = 0 and Y = 10 mi. It moves on a path for 10 hr such that x =’ t and y = 0.5t2 + 10, where t is in hours. An international fishing boundary is described by the line y = 2x +6.

a. Plot and label the path of the vessel and the boundary.

b. The perpendicular distance of the point XI, YI) from the line Ax + By + C =0 is given by

where the sign is chosen to make d ?: O. Use this result to plot the

distance of the fishing vessel from the fishing boundary as a function of time for 0 ::: t ::: 10 hr

**Sections 5.2 and 5.4**

9. Plot columns 2 and 3 of the following matrix A versus column 1. The data in column I is time (seconds). The data in columns 2 and 3 is force (newtons).

10. * Many engineering applications use the following “small angle”

approximation for the sine to obtain a simpler model that is easy to

understand and analyze. This approximation states that sin x :::x:,::w:here x must be in radians. Investigate the accuracy of this approximation by creating three plots. For the first, plot sin x and x versus x for 0 ::: x ::: I. For the second, plot the approximation error sin x – x versus x for 0::: x ::: 1. For the third, plot the relative error [sin(x) -x]/sin(x) versus x for 0 ::: x ::: I. How small must x be for the approximation to be accurate within 5 percent?

11. You can use trigonometric identities to simplify the equations that appear in many engineering applications. Confirm the identity tan(2x) = 2 tan x / (J – tarr’ x) by plotting both the left and the right sides versus x over the range 0 ::: x ::: Zn .

12. The complex number identity eix = cos x +i sin x is often used to convert the solutions of engineering design equations into a form that is relatively easy to visualize. Confirm this identity by plotting the imaginary part versus the real part for both the left and right sides over the range 0″. x- 2n

13. Use a plot over the range 0 ::: x ::: 5 to confirm that sinCix) = i sinh x.

14.* The function )”(t) = 1 – e”!”, where t is time and b > 0, describes many engineering processes, such as the height of liquid in a tank as it is being filled and the temperature of an object being heated. Investigate the effect of the parameter bony(t). To do this, plot y versus t for several values of b on the same plot. How long will it take for y(t) to reach 98 percent of its steady-state value?

I5. The following functions describe the oscillations in electrical circuits and the vibrations of machines and structures. Plot these functions on the same plot. Because they are similar, decide how best to plot and label them to avoid confusion.

16. The data for a tension test on a steel bar appears in the following table. The elongation is the change in the bar’s length. The bar was stretched beyond its elastic limit so that a permanent elongation remained after the tension force was removed. Plot the tension force versus the elongation. Be sure to label the parts of the curve that correspond to increasing and decreasing tension.

17. In certain kinds of structural vibrations, a periodic force acting on the structure will cause the vibration amplitude to repeatedly increase and decrease with time. This phenomenon, called beating. also occurs in musical sounds. A particular structure’s displacement is described by

where y is the displacement in inches and t is the time in seconds. Plot y versus t over the range 0 t 20 for II = 8 rad/see and 12 = I rad/sec. Be sure to choose enough points to obtain an accurate plot.

18.* The height h(t) and horizontal distance x(t) traveled by a ball thrown at an angle A with a speed v are given by

.

At Earth’s surface the acceleration due to gravity is g = 9.81 m/s”.

a. Suppose the ball is thrown with a velocity v = 10 mls at an angle of 35°. Use MATLAB to compute how high the ball will go, how far it

will go, and how long it will take to hit the ground.

b. Use the values of v and A given in part a to plot the ball’s trajectory; that is, plot h versus x for positive values of h.

c. Plot the trajectories for v = 10 mls corresponding to five values of the angle A: 20°, 30°, 45°, 60°, and 70°. d. Plot the trajectories for A = 45° corresponding to five values of the initial velocity u: 10, 12, lA, 16, and 18 mls.

19. The perfect gas law relates the pressure p, absolute temperature T, mass 111, and volume V of a gas. It states that

The constant R is the gas constant. The value of R for air is 286.7

N . m/kg . K. Suppose air is contained in a chamber at room temperature (20cC = 293 K). Create a plot having three curves of the gas pressure in N/m2 versus the container volume V in m3 for 20≤V ≤ 100. The hree curves correspond to the following masses of air in the container: In = I kg; m = 3 kg; and III = 7 kg.

20. Oscillations in mechanical structures and electric circuits can often be described by the function

**y(t) = e-I/r sin(wt +¢)**

where t is time and to is the oscillation frequency in radians per unit time. The oscillations have a period of 2Jf[ca, and-their amplitudes decay in time at a rate determined by T, which is called the time constant. The smaller T is, the faster the oscillations die out.

a. Use these facts to develop a criterion for choosing the spacing of

the t values and the upper limit on t to obtain an accurate plot of y(t).

(Hint: Consider two cases: 4T > 2Jf/w and 4T < 2Jf/w.)

b. Apply your criterion, and plot y(t) for T = 10, to = n , and ¢ = 2.

c. Apply your criterion, and plot y(t) for T = 0.1, co = 8Jf, and ¢ = 2.

21. When a constant voltage was applied to a certain motor initially at rest, its rotational speed s(t) versus time was measured. The data appears in the

**Section 5.3**

22. The following table shows the average temperature for each year in a certain city. Plot the data as a stem plot, a bar plot, and a stairs plot.

23. $10,000 invested at 5 percent interest compounded annuaIly wiIl grow according to the formula y

**y(k) = 1O\J.05)k**

where k is the number of years (k =0, 1, 2 … ). Plot the amount of money in the account for a 1O-year period. Do this problem with four types of plots: the xy plot, the stem plot, the stairs plot, and the bar plot.

24. The volume V and surface area A of a sphere of radius r are given by

**V = ~Jrr3 A = 4Jrr2**

a. Plot V and A versus r in two subplots, for 0.1 ::: r ::: 100 m. Choose

axes that will result in straight-line graphs for both V and A.

b. Plot V and r versus A in two subplots, for 1 ::: A ::: 104 m2. Choose

axes that will result in straight-line graphs for both V and r.

25. The current amount A of a principal P invested in a savings account

paying an annual interest rate r is given by

**A = P (I+ ~)”I**

. where 11 is the number of times per year the interest is compounded. For continuous compounding, A = Perl. Suppose $10,000 is initially invested at 3.5 percent (r =0.035).

a. Plot A versus t for 0 t 2’J years for four cases: continuous

compounding, annual compounding (n = I), quarterly compounding

12 = 4), and monthly compounding (n = 12). Show all four cases on

the same subplot and label each curve. On a second subplot, plot

the difference between the amount obtained from continuous

compounding and the other three cases. .

b. .Redo part a but plot A versus t on log-log and semilog plots. Which plot gives a straight line?

26. The grades of 80 students were distributed as follows

Use the pie chart function pie to plot the grade distribution. Add the

title “GradeDistribution” to the chart. Use the gtext function or the Plot Editor to add the letter grades to the sections of the pie chart.

27. Ifwe apply a sinusoidal voltage Vi to the circuit shown in Figure P27, then eventually the output voltage Vo will be sinusoidal also, with the same frequency, but with a different amplitude and shifted in time relative to the input voltage. Specifically, if Vi = Ai sin cot, then Vo = Ao sinew! + ¢». The frequency-response plot is a plot of Ao/ Ai versus frequency w. This ratio depends on w as follows:

**Ao I RCs I Ai = RCs + I**

where s·= toi. For RC = 0.1 s, obtain the log-log plot of lAo/Ad versus w over the range of frequencies I ~ to ~ 1000 rad/s. Compare the plot with Figure 5.3-5, which is for a similar circuit.

28. If we apply a sinusoidal voltage Vi to the circuit shown in Figure P28, then eventually the output voltage Vo will be sinusoidal also, with the same frequency, but with a different amplitude and shifted in time relative to theinput voltage. Specifically, if Vi = Ai sin tat , then Vn = An sin(wt + ¢). The frequency-response plot is a plot of Ani Ai versus frequency w. This ratio depends on w as follows

where s = coi . For R = 6 n, L = 3.6 X 10-3 H, and C = 10-6 F, plot

IAol Ai I versus to on rectilinear axes and on log-log axes over the range of frequencies 103 .:5to .:5 106 rad/s. Is there an advantage to using log-log axes?

**Section 5.5**

29. The distance a spring stretches from its “free length” is a function of how much tension force is applied to it. The following table gives the spring length y that the given applied force f produced in a particular spring. The spring’s free length is 4.7 in. Find a functional relation between f and x, . the extension from the free length (x = y – 4.7).

30.* In each of the following problems, determine the best function y(x) (linear, exponential, or power function) to describe the data. Plot the function on the same plot with the data. Label and format the plots appropriately.

31. The population data for a certain country is

Obtain a function that describes this data. Plot the function and the data on the same plot. Estimate when the population will be double its 1990 size.

32. * The half-life of a radioactive substance is the time it takes to decay by half. The half-life of carbon 14, which is used for dating previously living things, is 5500 years. When an organism dies, it stops accumulating carbon 14. The carbon 14 present at the time of death decays with time. Let C(t)/ C(O) be the fraction of carbon 14 remaining at time t. In radioactive carbon dating, scientists usually assume that the remaining fraction decays exponentially according to the following formula

a. Use the half-life of carbon 14 to find the value of the parameter b, and plot the function. b. If 90 percent of the original carbon 14 remains, estimate how long ago the organism died.

c. Suppose our estimate of b is off by ± I percent. How does this error affect the age estimate in b?

33. Quenching is the process of immersing a hot metal object in a bath for a specified time to obtain certain properties such as hardness. A copper sphere 25 mm in diameter, initially at 300oe, is immersed in a bath at O°c. The following table gives measurements of the sphere’s temperature versus time. Find a functional description of this data. Plot the function and the data on the same plot.

34. The useful life of a machine bearing depends on its operating temperaicre, as the following data shows. Obtain a functional description of this data. Plot the function and the data on the same plot. Estimate a bearing’s life if it operates at 150°F.

35. A certain electric circuit has a resistor and a capacitor. The capacitor is initially charged to 100 V. When the power supply is detached, the capacitor voltage decays with time, as the following data table shows. Find a functional description of the capacitor voltage v as a function of time t. Plot the function and the data on the same plot.

**Sections 5.6 and 5.7**

36. * The distance a spring stretches from its “free length” is a function of how much tension force is applied to it. The following table gives the spring length y that was produced in a particular spring by the given applied force f. The spring’s free length is 4.7 in. Find a functional relation between f and x, the extension from the free length (x = y – 4.7). The function must pass through the origin (x = O, f = 0).

37. The following data gives the drying time T of a certain paint as a function of the amount of a certain additive A.

Q. Find the first-, second-, third-, and fourth-degree polynomials that fit the data and plot each polynomial with the data. Determine the quality of the curve fit for each by computing J, S, and r2,

b. Use the polynomial giving the best fit to estimate the amount of

additive that minimizes the drying time

38. * The following data gives the stopping distance d as a function of initial speed v, for a certain car model. Find a quadratic polynomial that fits the data. Determine the quality of the curve fit by computing J, S, and r2.

39. If the acceleration a is constant, Newton’s law predicts that the distance travelled versus time is a quadratic function: d = ~r2 +hr. Thefollowing data was taken for a cyclist. Use the data to estimate the cyclist’s acceleration a.

40. * The number of twists y required to break a certain rod is a function of the percentage XI and X2 of each of two alloying elements present in the rod. The following table gives\some pertinent data. Use linear multiple regression to obtain a model y = ao + alxl +a2X2 of the relationship between the number of twists and the alloy percentages. In addition, find the maximum percent errdr in the predictions.

41. The following represents pressure samples, in pounds per square inch (psi), taken in a fuel line once every second for 10 sec.

a. Fit a first-degree polynomial, a second-degree polynomial, and a

third-degree polynomial to this data. Plot the curve fits along with the data points.

b. Use the results from part a to predict the pressure at t = II sec.,

Explain which curve fit gives the most reliable prediction. Consider

the coefficients of determination and the residuals for each fit in

making your decision.

42. A liquid boils when its vapor pressure equals the external pressure acting on the surface of the liquid. This is the reason why water boils at a lower temperature at higher altitudes. This information is important for chemical, nuclear, and other engineers who must design processes utilizing boiling liquids. Data on the vapor pressure P of water as a function of temperature T is given in the following table. From theory we know that In P is

proportional to liT. Obtain a curve fit for peT) from this data. Use the

fit to estimate the vapor pressure at 285 K and at 300 K.

43. The salt content of water in the environment affects living organisms and causes corrosion. Environmental and ocean engineers must be aware of these effects. The solubility of salt in water is a function of the water temperature. Let S represent the solubility of NaCI (sodium chloride) as grams of salt in 100 g of water. Let T be temperature in “C. Use the following data to obtain a curve fit for S as a function of T. Use the fit to estimate S when T = 25cC.

44. The amount of dissolved oxygen in water affects living organisms and chemical processes. Environmental and chemical engineers must be aware

of these effects. The solubility of oxygen in water is a function of the

water temperature. Let S represent the solubility of O2 as millimoles of O2 per liter of water. Let T be temperature in “C. Use the following data to obtain a curve fit for S as a function of T. Use the fit to estimate S when T = 8°C and T = 50cC

y(x) = al + a2lnx

45. The following function is linear in the parameters al and Q2.

y(x) = al + a2lnx Use least squares regression with the following data to estimate the values of al and a2. Use the curve fit to estimate the values of y at x = 1.5 and at

x = 11.

46. Chemical, environmental, and nuclear engineers must be able to predict the changes in chemical concentration in a reaction. A model used for many single reactant processes is:

Rate of change of concentration = =kC”

where C is the chemical concentration and k is the rate constant. The order of the reaction is the value of the exponent n. Solution methods for differential equations (which are discussed in Chapter 8) can show that the solution for a first-order reaction (n = 1) is

**C(l) = C(O)e-kl**

The following data describes the reaction

Use this data to obtain a least squares fit to estimate the value of k.

47. Chemical, environmental, and nuclear engineers must be able to predict the changes in chemical concentration in a reaction. A model used for many single reactant processes is:

Rate of change of c.onc.entration = -kCn

where C is the chemical concentration and k is the rate constant. The order of the reaction is the value of the exponent n. Solution methods for differential equations (which are discussed in Chapter 8) can show that the solution for a first-order reaction (n = I) is

**C(t) = C(O)e-kl**

and the solution for a second-order reaction (n = 2) is

**1 1
C(t) = C(O) +kt**

The following data from [Brown, 1994] describes the gas-phase

decomposition of nitrogen dioxide at 300°C.

**2N02 ~ 2NO +02
•
Time t (s) C (mol N01/L)
o 0.0100
50 0.0079
100 0.0065
200 0.0048
300 0.0038**

Determine whether this is a first-order or second-order reaction, and

estimate the value of the rate constant k.

48. Chemical, environmental, and nuclear engineers must be able to predict the changes in chemical concentration in a reaction. A model used for many single reactant processes is:

Rate of change of concentration = -kCn

where C is the chemical concentration and k is the rate constant. The order of the reaction is the value of the exponent II. Solution methods for

**Section 5.8**

49. The popular amusement ride known as the corkscrew has a helical shape. The parametric equations for a circular helix are

x = a cost

y =a sint

z = bt

where a is the radius of the helical path and b is a constant that determines the “tightness” of the path. In addition, if b > 0, the helix has the shape of a right-handed screw; if b < 0, the helix is left-handed. Obtain the three-dimensional plot of the helix for the following three cases and compare their appearance with one another. Use 0 ::: t ::: IOn

and a = 1.

a. b = 0.1

b. b = 0.2

c. b = -0.1

50. A robot rotates about its base at two revolutions per minute while lowering its arm and extending its hand. It lowers its arm at the rate of 120° per minute and extends its hand at the rate of 5 mlmin. The arm is 0.5 m long. The xyz coordinates of the hand are given by

where t is time in minutes.

Obtain the three-dimensional plot of the path of the hand for 0 ~ t ~

0.2 min. 51. Obtain the surface and contour plots for the function z = x2 – 2xy +4y2, showing the minimum at x = y = O.

52. Obtain the surface and contour plots for the function z =

_x2 +2xy +3y2. This surface has the shape of a saddle. At its

addlepoint at x = y = 0, the surface has zero slope, but this point does not correspond to either a minimum or a maximum. What type of contour . lines correspond to a saddlepoint?

53. Obtain the surface and contour plots for the function z =(x – y2)(x – 3y2). This surface has a singular point at x = y = 0, where the surface has zero lope, but this point does not correspond to either a minimum or a maximum. What type of contour lines correspond to a singular point?

54. A square metal plate is heated to 800e at the comer corresponding to.x = )’= I. The temperature distribution in the plate is described by T = 80e-(x-l)2 e-3(y-1f

Obtain the surface and contour plots for the temperature. Label each axis. hat is the temperature at the comer corresponding to x = y = O?

55. The following function describes oscillations in some mechanical

tructures and electric circuits:

z(t) = e-I r sin(wt +¢) In this function t is time, and co is the oscillation frequency in radians per unit time. The oscillations have a period of 2JT[t», and their amplitudes decay in time at a rate determined by r , which is called the time constant. The smaller r is, the faster the oscillations die out. Suppose that ¢ = 0, to = 2, and r can have values in the range 0.5 r 10 sec. Then the preceding equation becomes z(t) =-.e-I/T sin(2r)~’ ‘.

Obtain a surface plot and a contour plot of this function to help visualize the effect of r for 0 ::::t ::::15 sec. Let the x variable be time t and the y variable be r , 56. Many applications require us to know the temperature distribution in an object. For example, this information is important for controlling the material properties such as hardness, when cooling an object formed from molten metal. In a heat transfer course the following description of the temperature distribution in a flat rectangular metal plate is often derived.

The temperature on three sides is held constant at T” and at T2 on the fourth side (see Figure P56). The temperature T(x, y) as a function of the xy coordinates shown is given by

The given data for this problem are: T, = 70°F, T2 = 200″F, and W =

L = 2 ft. Using a spacing of 0.2 for both x and y, generate a surface mesh plot and a contour plot of the temperature distribution.

57. The electric potential field V at a point, due to two charged particles, is given by

where q, and q2 are the charges of the particles in Coulombs (C), r, and ri are the distances of the charges from the point (in meters), and Eo is the permittivity of free space, whose value is

Eo = 8.854 X IO-J2 C2/N . m2

Suppose the charges are ql = 2 X 10-10 C and q2 = 4 X 10-10 C. Their respective locations in the xy plane are (0.3, 0) and (-0.3,0) m. Plot the electric potential field on a 3D surface plot with V plcitted on the z-axis over the ranges -0.25 ~ x ~ 0.25 and -0.25 ~ y ~ 0.25. Create the plot two ways: Q. by using the surf function and b. by using the meshe function.

58. The grades of 80 students were distributed as follows.

Use the 3D pie chart function pie3 to plot the grade distribution.

Add the title “Grade Distribution” to the chart. Use the Plot Editor to add the letter grades to the sections of the pie chart.

59. Refer to Problem 22 of Chapter 4. Use the function file created for that problem to generate a surface mesh plot and a contour plot of x versus h and W for 0 ~ W ~ 500 N and for 0 ~ h ~ 2 m. Use the values: kJ = 104 N/m; k2 = 1.5 X 104 N/m; and d = 0.1 m.

60. Refer to Problem 25 of Chapter 4. To see how sensitive the cost is to location of the distribution center, obtain a surface plot and a contour plot of the total cost as a function of the x and y coordinates of the distribution center location. How much would the cost increase if we located the center 1 mi in any direction from the optimal location?

61. Refer to Example 3.2-2 of Chapter 3. Use a surface plot and a contour plot of the perimeter length L as a function of d and e over the ranges1 d 30 ft and 0.1 e 1.5 rad. Are there valleys other than the one corresponding to d =7.5984 and e = 1.0472? Are there any saddle points?