Final Exam - Answers.

Math For Social Sciences. Practice Exam

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.

$\begin{eqnarray*}3x+y&=&-3\\ -4x+2y&=&1 \end{eqnarray*}$

Answer: $\displaystyle x=-\frac{7}{10}$ , $\displaystyle y=-\frac{9}{10}$ .

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:

$\displaystyle 12(10^{1-x})=60$ .

Answer: $\displaystyle x=-\frac{\ln\,5}{\ln\,10}+1\approx.301029$ .
$\displaystyle \log_{7}(2x+1)=1+\log_{7}(x-2)$

Answer: x=3.

7.

Which of the following (if any) represent y as a function of x?

(a) $\displaystyle y=\sqrt{x-1}$ .

Answer: This represents a function.
(b) x=(y-1)²

Answer: This does not represent a function.

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:

$\begin{displaymath}\begin{array}{rrrrrrr} x&+&y&-&4z&=&0\\ 2x&+&y&-&3z&=&2 \end{array}.\end{displaymath}$

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 $\displaystyle 6\frac{1}{2}$ 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 $\displaystyle f(x)=3^{x}-1$ .

What is the domain of f(x).

Answer: The domain is all real numbers, or in interval notation, $(-\infty,\,\infty)$ .
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 $\displaystyle \log_{b}2=0.2361$ , $\displaystyle \log_{b}3=0.3122$ , and $\displaystyle \log_{b}5=0.3708$ .
Find $\displaystyle \log_{b}(24b)$ .

Answer: $\displaystyle \log_{b}(24b)$ = $\displaystyle \log_{b}b+3\,\log_{b}2+\log_{b}3$ =2.0205.

12.

Using the Gauss-Jordan method, solve the system of equations:

$\begin{displaymath}\begin{array}{rrrrrrr} x&-&2y&+&4z&=&6\\ x&+&y&+&13z&=&6\\ -2x&+&6y&-&z&=&-10 \end{array}.\end{displaymath}$

Answer: $x=-14,\,y=-6,\,z=2$ .

14.

A patient was admitted to a hospital with a declining white blood cell count, described by the equation:

y=t²-30t+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:

$\begin{displaymath}\begin{array}{rrrrrrr} 24a&+&30b&+&12c&=&306\\ 4a&+&5b&+&3c&=&59\\ 15a&+&18b&+&9c&=&201 \end{array}.\end{displaymath}$

16.

Let $\displaystyle A=\left[\begin{array}{rr} 5&0\\ -1&3\\ 4&7 \end{array}\right]$ $\displaystyle B=\left[\begin{array}{r} 6\\ 1\\ 0 \end{array}\right]$ $\displaystyle C=[\begin{array}{rrr} 1&3&-4\end{array}]$ $\displaystyle D=\left[\begin{array}{rr} -1&4\\ 3&7 \end{array}\right]$

$\displaystyle E=\left[\begin{array}{rr} 2&5\\ 1&6 \end{array}\right]$

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:
  
  $\begin{displaymath}\left[\begin{array}{rr} 4&-3\\ -5&-8 \end{array}\right].\end{displaymath}$
- BA
  
  Answer: Not possible.
- CB
  
  Answer: [9]

Formulas

Future value for simple interest:

A=P(1+rt).
Future value for compound interest:

$\begin{displaymath}A=P(1+i)^{n}\;\;\;\mbox{ or }\;\;\;A=P\left(1+\frac{r}{n}\right)^{nt}.\end{displaymath}$
Future value of an ordinary annuity:

$\begin{displaymath}S=\frac{R=[(1+i)^{n}-1]}{i}.\end{displaymath}$
Future value of an annuity due:

$\begin{displaymath}S=\frac{R[(1+i)^{n+1}-1]}{i}-R.\end{displaymath}$
Present value of an annuity:

$\begin{displaymath}P=\frac{R[1-(1+i)^{-n}]}{i}.\end{displaymath}$
Amortizing a loan:

$\begin{displaymath}R=\frac{Pi}{1-(1+i)^{-n}}.\end{displaymath}$

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June 2, 1998