1. In this project, we solve cubic equations of the form
x3+mx=n
Note that there is no quadratic term. This special form was first solved by the Italian mathematicians Scipione del Ferro and Niccolò Fontana Tartaglia early in the sixteenth century. Tartaglia revealed the secret to solving the special cubic equation in a poem. He first found values uand vto satisfy the system
u−v=n uv=m
3
3
a.We will use Tartaglia’s method to solve
x3+6x=7 What are the values of mand n?
b.Substitute the values of mand ninto Tartaglia’s system, then use substitution to solve for uand v. You should find two possible solutions.
c. For each solution of the system, compute
x=√3 u−√3 v. You should get the same value of x
for each (u,v).
d.Check that your value for xis a solution of
x3+6x =7.
2. Tartaglia’s method always works to solve the special cubic equation, even when uand vare not convenient values. We will show why in this project.
a.Expand the expression (a−b)3+3ab(a−b)and complete the identity.
(a−b)3+3ab(a−b)=____________
b.Your answer to part (a) is actually Tartaglia’s special cubic in disguise. Substitute x=a−b,m=3ab,
and n=a3−b3to see this. Therefore, if we can find numbers aand bthat satisfy
3ab=m a3−b3=n
then the solution to Tartaglia’s cubic is x=a−b.
c. Compare the system in part (b) to the system from Project 1,
u−v=n uv=m
3
3
to show that u=a3and v=b3.
d.Use your answer to part (c) to show that Tartaglia’s value, x=√3 u−√3 v, is a solution of x3+mx=n.
3. Use Tartaglia’s method to solve the equation
x3+3x=2 by carrying out the following steps.
a.Identify the values of mand nfrom Equation (1) and write two equations for uand v.
b.Solve for values of uand v. You will need to use the quadratic formula.
c. Take the positive values of uand v. Write the solution
x=√3 u−√3 v. Do not try to simply the radical
expression; instead, use your calculator to check the solution numerically.
4. We can solve any cubic equation by first using a
substitution to put the equation in Tartaglia’s special form.
a.Consider the equation X3+b X2+c X+d=0. Make the substitution X =x−b3, and expand the left side of the equation.
b.What is the coefficient of x2in the resulting equation? What are the values of mand n?
c. If you solve the special form in part (a) for x, how can you find the value of Xthat solves the original equation?
81. V =C 1− t n , for n 82. r= dc 1−ec, for c 83. p q = r q+r, for q 84. I = E R+r n , for R
5. The time, T, it takes for the Moon to eclipse the Sun totally is given (in minutes) by the formula
T =1 v r D R −d
where dis the diameter of the Moon, Dis the diameter of the Sun, ris the distance from the Earth to the Moon,
Ris the distance from the Earth to the Sun, and vis the speed of the Moon.
a.Solve the formula for vin terms of the other variables.
b.It takes 2.68 minutes for the Moon to eclipse the Sun. Calculate the speed of the Moon given the following values:
d=3.48×103km D=1.41×106km
r =3.82×105km R=1.48×108km
6. The stopping distance, s, for a car traveling at speed
vmeters per second is given (in meters) by
s=vT + v
2
2a
where Tis the reaction time of the driver and ais the average deceleration as the car brakes. Suppose that all the cars on a crowded motorway maintain the appropriate spacing determined by the stopping distance for their speed. What speed allows the maximum flow of cars along the road per unit time? Using the formula time=distancespeed , we see that the time interval, t, between cars is
t =s v+
L v
where Lis the length of the car. To achieve the
maximum flow of cars, we would like tto be as small as possible. (Source: Bolton, 1974)
a.Substitute the expression for sinto the formula for t, then simplify.
b.A typical reaction time is T =0.7 seconds, a typical car length is L=5 meters, and a=7.5 meters per second squared. With these values, graph tas a function of vin the window
Xmin=0 Xmax=20 Ymin=0 Ymax=3
c. To one decimal place, what value of vgives the minimum value of t? Convert your answer to miles per hour.
7. Many endangered species have fewer than 1000 individuals left. To preserve the species, captive
breeding programs must maintain a certain effective population, N, given by
N = 4F M F+M
where Fis the number of breeding females and Mthe number of breeding males. (Source: Chapman and Reiss, 1992)
a.What is the effective population if there are equal numbers of breeding males and females?
b.In 1972, a breeding program for Speke’s gazelle was established with just three female gazelle. Graph the effective population, N, as a function of the number of males.
c. What is the largest effective population that can be created with three females? How many males are needed to achieve the maximum value?
d.With three females, for what value of Mis N =M?
e. The breeding program for Speke’s gazelle began with only one male. What was the effective population?
8. When a drug or chemical is injected into a patient, biological processes begin removing that substance. If no more of the substance in introduced, the body removes a fixed fraction of the substance each hour. The amount of substance remaining in the body at time tis an exponential decay function, so there is a biological half-lifeto the substance denoted by Tb. If the substance is a radioisotope, it undergoes radioactive decay and so has a physical half-life as well, denoted Tp. The effective
half-life,denoted by Te, is related to the biological and physical half-lives by the equation
1 Te = 1 Tb + 1 Tp
The radioisotope 131I is used as a label for the human
serum albumin. The physical half-life of 131I is 8 days.
(Source: Pope, 1989)
a.If 131I is cleared from the body with a half-life of
21 days, what is the effective half-life of 131I?
b.The biological half-life of a substance varies considerably from person to person. If the biological half-life of 131I is xdays, what is the effective half
life?
c.Let f(x)represent the effective half-life of 131I when
the biological half-life is xdays. Graph y= f(x).
d.What would the biological half-life of 131I need to be
to produce an effective half-life of 6 days? Label the corresponding point on your graph.
e.For what possible biological half-lives of 131I will the
9. Animals spend most of their time hunting or foraging for food to keep themselves alive. Knowing the rate at which an animal (or population of animals) eats can help us determine its metabolic rate or its impact on its habitat. The rate of eating is proportional to the availability of food in the area, but it has an upper limit imposed by mechanical considerations, such as how long it takes the animal to capture and ingest its prey. (Source: Burton, 1998)
a.Sketch a graph of eating rate as a function of quantity of available food. This will be a qualitative graph only; you do not have enough information to put scales on the axes.
b.Suppose that the rate at which an animal catches its prey is proportional to the number of prey available, or rc=ax, where ais a constant and xis the number of available prey. The rate at which it handles and eats the prey is constant, rh =b. Write expressions for Tcand Th, the times for catching and handling
Nprey.
c. Show that the rate of food consumption is given by
y= abx b+ax =
bx b/a+x
Hint: y= NT,where Nis the number of prey consumed in a time interval T, where T =Tc+Th.
d.In a study of ladybirds, it was discovered that larvae in their second stage of development consumed aphids at a rate y2aphids per day, given by
y2=
20x x+16
where xis the number of aphids available. Larvae in the third stage ate at rate y3, given by
y3=
90x x+79
Graph both of these functions on the domain 0≤x≤140.
e. What is the maximum rate at which ladybird larvae in each stage of development can consume aphids?
10. A person will float in fresh water if his or her density is less than or equal to 1 kilogram per liter, the density of water. (Density is given by the formula density= volumeweight.) Suppose a swimmer weighs 50+F kilograms, where F
is the amount of fat her body contains. (Source: Burton, 1998)
a.Calculate the volume of her nonfat body mass if its density is 1.1 kilograms per liter.
b.Calculate the volume of the fat if its density is 0.901 kilograms per liter.
c.The swimmer’s lungs hold 2.6 liters of air. Write an expression for the total volume of her body, including the air in her lungs.
d.Write an expression for the density of the swimmer’s body.
e.Write an equation for the amount of fat needed for the swimmer to float in fresh water.
f. Solve your equation. What percent of the swimmer’s weight is fat?
g.Suppose the swimmer’s lungs can hold 4.6 liters of air. What percent body fat does she need to be buoyant?