- 1.
- Find an equation for the line through the point (-2,5), which is
perpendicular to the line 3y-x+2=0.
Answer: The equation is y=-3x-1.
- 2.
- The cost of producing x-units of a product is given by:
C(x)=20x+100. The product sells for $45 per unit.
- (a) Find the break-even point.
Answer: The break-even point is 4 units.
- What revenue will the company receive if it sells just the
number of units from part (a)?
Answer: The revenue is $180.
- Find the profit function, P(x).
Answer: The profit function is
P(x)=25x-100.
- Find the average cost per unit to produce 100 units.
Answer: The average cost per unit is $21.
- 3.
- A 45-year old man puts $1,500 in a retirement account at the
end of the each quarter until he reaches the age of 60 and makes not
further deposits. If the account pays 12% interest compounded
quarterly, how much will be in the account when the man retires at age
65?
Answer: The amount of money at the end will be about $441,739.
- 4.
- Use the method of matrix inverses to solve the system of
equations.
Answer:
,
.
- 5.
- Sketch the graph of
f(x)=|3x-12|. Identify and include any
intercepts in your sketch.
Answer:
The x-intercept is 4, and the y-intercept is 12.
- 6.
- Solve each of the following for x:
- 7.
- Which of the following (if any) represent y as a function of
x?
Explain your answers to 7a and 7b. That is, if you believe that 7a is
(or is not) a function, explain how you came to this conclusion. do
the same for 7b.
Answer: The first equation represents a function, because its graph passes
the vertical line test.
The second equation is not a function, because for many values of
x in the domain, there will be two different values for y in the
range, one positive and the other negative. This violates the
definition of a function.
- 8.
- Use the elimination method to solve the following system:
Answer: (Since there are only two equations and three unknowns,
the solutions will be parametrized in terms of one variable)
All solutions are given by (x, 8-5x, 2-x),
where x is arbitrary.
There are other parametrizations.
- 9.
- Consider the supply and demand curves in the sketch below:
- At what price are 20 items supplied?
Answer: 20 items are supplied at $140.
- Find the equilibrium demand and the equilibrium price.
Answer: The equilibrium demand is 10 units, and the equilibrium
price is $120 per unit.
- Are the supply and demand curves linear, quadratic, exponential,
logarithmic, or none of these?
Answer: The supply and demand curves are both linear.
- 10.
- A small resort must add a swimming pool to compete with a new
resort built nearby. The pool will cost $28,000. The resort borrows
the money and agrees to repay it with equal payments at the end of
each quarter for
years at an interest rate of 6%
compounded quarterly. Find the amount of each payment.
Answer: The amount of each payment is $1,308.49.
- 11.
- Consider the function
.
- What is the domain of f(x).
Answer: The domain is all real numbers, or in interval notation,
.
- Find any intercepts and asymptotes.
Answer: There is a horizontal asymptote at y=-1.
- Sketch the graph of f(x), including the intercepts and
asymptotes in your sketch.
Answer:
- If
,
,
and
.
Find
.
Answer:
=
=2.0205.
- 12.
- Using the Gauss-Jordan method, solve the system of
equations:
Answer:
.
- 14.
- A patient was admitted to a hospital with a declining white
blood cell count, described by the equation:
y=t^{2}-30t+2225
where t is time in hours since the patient was admitted, and y is
the blood count.
- What was the patient's minimum blood count, and when did it
occur?
Answer: The patient's minimum blood count was 2,000, and it
occurred 15 hours after admission.
- When the patient's blood count reached 3225, the patient was
discharged. When did this occur?
Answer: The patient's blood count reached 3223 and was
discharged 50 hours after admission.
- Is the given equation linear, quadratic, exponential,
logarithmic, or none of these?
Answer: The given equation is quadratic.
- 15.
- The Waputi Indians make woven blankets, rugs, and skirts. Each
blanket requires 24 hrs for spinning the yarn, 4 hrs for dying the
yarn, and 15 hrs for weaving. Rugs require 30 hrs for spinning, 5 hrs
for dying, and 18 hrs for weaving. Skirts require 12 hrs for
spinning, 3 hrs for dying, and 9 hrs for weaving. There are 306 hours
available for spinning, 59 hrs for dying, and 201 hrs for weaving.
Set up a system of equations which could be used to determine the
number of blankets, rugs, and skirts made. Do not solve
Answer:
Let a= the number of blankets, b= the number of rugs and
let c= the number of skirts. The system of equations is:
- 16.
- Let
- Give the dimensions of each matrix, and identify any square,
column, or row matrices.
Answer: The dimensions are: For A, 3 x 2. For B, 3 x 1, and
the matrix is a column matrix. For C, 1 x 3, and the matrix is a row
matrix. For D, 2 x 2, and the matrix is a square matrix. For E, 2 x
2, and the matrix is a square matrix.
- Find the following if possible:
- CA
Answer: [-14,-19],
- E-2D
Answer:
- BA
Answer: Not possible.
- CB
Answer: [9]
Formulas
- Future value for simple interest:
A=P(1+rt).
- Future value for compound interest:
- Future value of an ordinary annuity:
- Future value of an annuity due:
- Present value of an annuity:
- Amortizing a loan: