 Precalculus Mathematics I Exam 2

The questions on the exam were as follows:

Answer all questions carefully. Do all steps with brief explanations similar to what was done in the examples in the notes. I prefer that you send your exams electronically. Your exams need not be typed. Follow this procedure:
1) On the course home page, click on Exam Dropboxes folder
2) Click on Exam II dropbox
3) Click Add Attachments
4) Click My Computer icon on the right to browse for a file to attach and select your exam file
5) Click Open then OK
6) You will see the attachment appear below the submission box
7) Click Submit to send it to the instructor
Before you do #3 read the note at the end of the exam.

1. (10 points) 2x + y = 7
-2x + 3y = 5

2. a. A sweater cost a dealer \$32.00. At what price should he mark the sweater so he can give a discount of 25% and still make a mark-up profit of 20%? What is the final price if there is a 5% sales tax?

b. Two people set out simultaneously from two locations 12 miles apart and walk toward each other. One person walks 5 miles an hour faster than the other. Find the rate of speed of each person if they meet in one hour and ten minutes.

3. x^2 + x - 30 = 0

(b) Determine the vertex of the parabola y = x2 + x – 30.
(c) Sketch the graph of y = x2 + x – 30 using the information you found in parts a and b.

4. (10 points) How high will a baseball that is thrown up into the air at 48 feet per second go if it starts its flight at a height 10 feet above the ground?

5. (25 points)
(a) Solve -5 <= 3x + 1 <= 3 for x. Express your answer using a line graph and in interval notation.

Solve the following inequalities. Express your answer using interval notation

(b) A person has two choices for his salary. Plan A. A salary of \$900 per month plus a commission of 10% of sales OR Plan B A salary of \$1,200 per month plus a commission of 15% of sales in excess of \$8,000. For what amount of monthly sales is plan B better than plan A?

(c) Solve |x – 3| = 4 for x
(d) Sketch the graph of the function y = | x - 3 | and use it to verify your solution
of part c.

6. (10 points)
Use the quadratic formula to determine the roots of the equation -x2 + 4x + 6 = 0 to the nearest tenth.

7. (10 points) A rain gutter is to be made up of rectangular aluminum sheets 12 inches wide by turning up the sides edges 90 degrees. What depth (of the edges) will provide a maximum cross sectional area and thereby provide for the greatest flow of water? (Hint, think of quadratic equations and that you want to maximize the area.)

Extra.

Find the maximum or minimum value of the given quadratic function on the interval that is specified.

Minimize f(x) = x^2- 1 Interval = [-1, 3]

Note:

Read sections 3.2, 3.3 and 3.5 carefully. These (and other) sections illustrate the power of the quadratic function in solving a variety of problems. To understand this function you must be able to graph it.
Some key thoughts:
(a) Any function of the form y = ax2 + bx + c where a 0 is a quadratic.
(b) The formula x = gives you the x value of the vertex. Substitute this value in the equation (of part) to find the y value.

(c) To sketch the graph of a quadratic one frequently need only know the vertex (x and y values) and the roots (found by solving ax2 + bx + c = 0 by factoring or by using the quadratic formula).
(d) The quadratic formula is:

The answers to the first two questions are listed below. To see the full solutions for the exam please go ahead and purchase the solution set.

1.
2x + y = 7
-2x + 3y = 5
We add the 2 equations side by side to give:
2x – 2x + y + 3y = 7 + 5
4y = 12 // apply distributive law on x-terms and y-terms separately
y = 3 // divide both sides by 4
Next, we substitute y = 3 into the first equation as follows:
2x + 3 = 7
2x = 4 // add -3 to both sides and simplify the terms
x = 2 // divide both sides by 2
In conclusion, the solution is:
x = 2
y = 3

2.
a,
We have the formula:
Markup = (Sale Price / Cost) – 1
We rearrange the formula to derive:
Markup + 1 = Sale Price / Cost // add 1 to both sides
Sale Price = (Markup + 1) · Cost // multiply both sides with Cost
Since the markup is 20% or 0.2, and the cost is \$32.00, we can easily compute the sale price:
Sale Price = (0.2 + 1) · (\$32.00)
Sale Price = 1.2 · (\$32.00) // just take the sum of 1 and 0.2
Sale Price = \$38.40 // get the product
However, this is just the sale price after a discount of 25% is applied. In other words, \$38.4 is equal to 75% or 0.75 of the pre-discounted sale price. This means that the pre-discounted sale price of the sweater should be \$38.40 / 0.75 = \$51.20.
The tax amount is (5%) · (\$51.20) = \$2.56.
Therefore, the final sale price is given by \$51.20 + \$2.56 = \$53.76.
b,
Let’s assume that the person A walks 5 mph faster than the person B. If B moves at the rate of x mph, then A moves at (x + 5) mph.
Since they walk in opposite directions, their relative speed toward each other is equal to the sum of 2 speeds, that is (2x + 5) mph.
The distance is 12 miles, and the traveling time is 1 and 1/6 hours. So, we have the equation:
12 = (2x + 5) (1 + 1/6)
12 = (2x + 5) (7/6) // add 1 and 1/6 together
72 = 7 (2x + 5) // multiply both sides with 6
72 = 14x + 35 // expand the right side using distributive law
37 = 14x // add -35 to both sides
x = 37/14 // divide both sides by 14
x ˜ 2.64 // take the approximate value
That is the speed of the person B.
So, the speed of the person A equals 2.64 + 5 = 7.64 (mph).
In conclusion, their speeds are 2.64 mph and 7.64 mph.

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Subjects: Mathematics -> Algebra
Topic: Take home exam for Precalculus Mathematics
Level: College / University
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Take home, take-home, exam solutions, Precalculus Mathematics

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Project Details
Subjects: Mathematics -> Algebra
Topic: Take home exam for Precalculus Mathematics
Level: College / University
Tags:

Take home, take-home, exam solutions, Precalculus Mathematics

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