MATH 209 — FINAL EXAM
This is an open book quiz – feel free to use all the material at your disposal.
Please remember to show your methodology as well as the answer. How you solve the problems counts almost as much as your correct answer – in other words, even if you get the wrong answer, partial credit is available if you can show that you know how to approach the problem!
These are not difficult – just remember to follow logical rules
AND PLEASE BE CAREFUL — CHECK YOUR ARITHMETIC
WATCH OUT FOR THOSE QUESTIONS THAT NEED ± SOLUTIONS
There are 20 questions.
For questions 1 through 5, perform the indicated operations. If possible, simplify the answer.
1. [2 – 5(6 – 2)²]÷(8 – 4•3)
parens
[2-5(4)^2]/(8-12)
Powers
[2-5(16)]/-4
multiplication
(2-80)/-4
parens
-78/-4
Cancel the minus
78/4
simplify
39/2
2.
Expand:
27x^6 * 2x^-3 / 6x^3
Simplify:
54x^3 / 6x^3
Cancel x^3:
54/6
Simplify:
9
3.
Common denominators:
3x/x + 3/x
------
4x/x + 4/x
Simplify:
(3x+3)/x
------
(4x+4)/x
Cancel the x:
(3x+3)/(4x+4)
Factor:
3(x+1)/4(x+1)
Cancel the x+1:
3/4
4.
Factor:
(y-3)(y-1) / (y-3)(y+3)
------
(y+6)(y+1) / (y+6)(y-3)
Flip the second fraction and multiply:
(y-3)(y-1)(y+6)(y-3)
------
(y-3)(y+3)(y+6)(y+1)
Cancel like terms:
(y-3)(y-1)
------
(y+3)(y+1)
5.
= 1 / 16^3/4
= 1 / 2^3
= 1/8
6.Solve the equation: T = (3a + b)/2 for a.
Multiply by 2:
2T = 3a+b
Subtract b:
3a = 2T-b
Divide by 3:
a = (2T-b)/3
7.Factor completely: x³ + 1
Sum of cubes:
(x+1)(x^2-x+1)
For problems 8 – 11, solve each equation, inequality or system of equations. Choose whatever method you like, but remember to show your methodology along with your answer.
8.(x + 2)/3 = (x – 1)/4
Cross multiply:
3(x-1) = 4(x+2)
Distribute:
3x-3 = 4x+8
Subtract 3x:
-3 = x + 8
Subtract 8:
X = -11
9.
Square both sides:
x-1 = x^2-2x+1
Subtract x-1:
0 = x^2-3x+2
Factor:
0 = (x-2)(x-1)
X = 2 or 1
10.(7x –6)² = 48
Square root:
7x-6 = ± √48
Add 6:
7x = 6 ± √48
Divide by 7:
x = (6 ± √48)/7 = (6 ± 4√3)/7
11.a) 170 is what percent of 20?
170/20 = 8.5 = 850%
b) 20 is what percent of 170?
20/170 = about 0.117647 = about 11.7647%
12.Select the graph that best satisfies the equation:y = x² – 2x – 3
U shape, through (-1, 0) and (3, 0)
(c)
13. Select the equation that best fits the following graph:
Choice (d): y = (x/3)(|x – 4|), since it has a cusp
14.Write a quadratic equation for which the following are solutions for x: 7 and – 3,
(x-7)(x+3)
x^2-4x-21
15.Write a quadratic equation for which the following are solutions for x: ± 3i.
(x-3i)(x+3i)
x^2+9
Questions 16 through 20 are word problems. Take your time to make sure that your equations fit the words before you start solving them.
16.Hector can shovel show off the sidewalks in three hours (He doesn’t live in Southern California, of course). Oliver can shovel snow off the same sidewalks in 2 hours. How long will it take them to shovel the snow off
the sidewalks if they work together?
1/(1/3+1/2)
= 1/(5/6)
= 6/5
6/5 hours = 72 minutes
17.Park rangers catch, tag and then release 140 elk back into a National Park. One month later, they’re able to round up 130 elk, and find that 91 of them have been tagged. Assuming the ratio of tagged elk to the total in the sample that was rounded up is the same as the ratio of all tagged elk to the total elk population in the park, how many elk are in the park? By the way, they’re beautiful animals. I’ve seen them at the Grand Canyon.
91/130 = 140/x
Cross multiply:
91x = 130*140
X = 130*140/91
X = 200
200 elk
18.Staying on the animal theme, a tourist visits the zoo, and is interested in knowing how many birds and lions are on display. The Zookeeper tells him that between birds and lions there are 30 heads and 100 feet. From this information, can you tell how many birds and lions there are? If so, how many birds and lions are in this zoo?
Total animals:
B+L = 30
Total feet:
2B+4L=100
Multiply the first one by -2:
-2B-2L=-60
Add that to the second one:
2L = 40
L = 20
Get B:
B+20 = 30
B = 10
10 birds, 20 lions
19.In a triangle, the measure of the second angle is 40º less than the measure of the first angle. The measure of the third angle is twice the first. What are the measures of each of the angles in the triangle?
x + (x-40) + 2x = 180
4x-40 = 180
4x = 220
x = 55
So:
x-40 = 15
2x = 110
Angles: 55, 15, 110
20.A school with 125 teachers and 4650 students wants to maintain a student-to-teacher ratio of 30:1. How many teachers do they need to hire in order to meet that ratio?
4650/(125+x) = 30/1
4650 = 30(125+x)
155 = 125 + x
X = 30
30 teachers