## 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.

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: • . Answer: . • Answer: x=3. 7. Which of the following (if any) represent y as a function of x? • (a) . 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: 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.

• If , , and .
Find .

12.
Using the Gauss-Jordan method, solve the system of equations:

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?

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

• E-2D

• BA

• CB

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:

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June 2, 1998