Algebra 2 Part 2 Final Exam: Detailed Solutions and Explanations

This document presents a comprehensive set of solutions for an Algebra 2 Part 2 final exam. The solutions cover a range of topics, including exponential growth and decay, logarithmic equations, solving for earthquake intensity, analyzing sales projections, solving for the population of plant species, calculating expected values, and interpreting probabilities from surveys. Additionally, the solutions address trigonometric functions, unit circle applications, trigonometric identities, and solving trigonometric equations. Each problem is solved with detailed, step-by-step explanations and justifications, ensuring a thorough understanding of the concepts and methodologies used. This resource is ideal for students seeking to review and understand the solutions to their Algebra 2 final exam, providing a valuable tool for exam preparation and concept reinforcement.

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
1. Bradley dropped a ball from a roof 16 feet high. Each time the ball hits the ground, it bounces
the previous height. Find the height the ball will bounce after hitting the ground the fourth
time.
(SHOW WORK)
The height of the ball can be expressed by the formula y= 3
5 x
Where y is the umber of time the ball hits the ground and x the previous height.
1 st hit := 3
5∗16=9.6
2 nd hit : 3
5∗9.6=5.76
3 rd hit :=3
5∗5.76=3.456
4 th hit := 3
5∗3.456=2.0736
The height of the ball will be 2.0736 feet
2. The 2002 Denali earthquake in Alaska had a Richter scale magnitude of 6.7. The 2003 Rat
Islands earthquake in Alaska had a Richter scale magnitude of 7.8.
(SHOW WORK)
Suppose an architect has designed a building strong enough to withstand an earthquake 70 times
as intense as the Denali quake and 30 times as intense as the Rat Islands quake. Find which
structure is strongest. Explain your finding.

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
(SHOW WORK)
2002 earthquake scale 6.7
2003 earthquake scale 7.8
Building can withstand 70∗6.7=469∧30∗7.8=234. The structure that can withstand an
earthquake as intense as 70 times the Denali earthquake is the stronger structure.
3. Assume that a company sold 5.75 million motorcycles and 3.5 million cars in the year 2010.
The growth in the sale of motorcycles is 16% every year and that of cars is 25% every year. Find
when the sale of cars will be more then the sale of motorcycles.
(SHOW WORK)
Equation of motorcycles : y=5.75∗( 1.16 ) t
Equation of cars : y=3.5 ( 1.25 )t
Now finding when the sale of cars and motorcycles will be equal
5.75∗( 1.16 ) t =3.5 ( 1.25 ) t
5.75
3.5 = ( 1.25 )t
( 1.16 )t
23
14 = ( 125
116 )
t
ln ( 23
14 )=tln(125
116 )
t=
ln ( 23
14 )
ln ( 125
116 )=6.6436 years
After 6.6436 years the sale of cars will exceed that of motorcycles. This will be after 2016.
Assuming the current record were collected at 31st Dec 2010. The sale of cars will exceed that of
motorcycles from 22nd July, 2016 onwards.

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
In a survey in 2010, the population of two plant species were found to be growing exponentially.
Their growth is given by these equations: species A, and species B, ,
where t = 0 in the year 2010.
4. After how many years will the population of species A be equal to the population of species B
in the forest?
(SHOW WORK)
Equivalent:
P . of A=P of B iff :2000 e0.05 t =5000 e0.02t
2000
5000 = e0.02 t
e0.05 t
ln ( 2
5 )=0.02 t−0.05 t
ln (2
5 )=−0.03 t
t=30.54 yrs
t=30 yrs 6 months 15 days
5. A raffle prize of dollars is to be divided among 7x people. Write an expression for the
amount of money that each person will receive.
(SHOW WORK)
Each person will receive
14 x2
15
7 x = 14 x2
105 x = 14 x
105
¿ 2
15 x

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
6. The pressure exerted on the walls of a container by a gas enclosed within it is directly
proportional to the temperature of the gas. If the pressure is 6 pounds per square inch when the
temperature is find the pressure exerted when the temperature of the gas is
(SHOW WORK)
P=kT
6=k 440
k = 6
440 = 3
220
p= 3
220∗380=57
11 =5.1818 pounds
7. Blake and Ned work for a home remodeling business. They are putting the final touches on a
home they renovated. Working alone, Blake can paint one room in 9 hours. Ned can paint the
same room in 6 hours. How long will it take them to paint the room if they work together?
(SHOW WORK)
Blake: 1 room∈9 hrs
Ned: 1 room∈6 hrs
In 1 hrs: Blake 1
9 ∧Ned 1
6
After an hour they will have painted a total of 1
6 + 1
9 = 5
18
Now the whole room
18
18∗18
5 ∗1=3.6 hrs
Working together they will take 3.6 hrs. to paint the whole room.

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
8. A tourist boat is used for sightseeing in a nearby river. The boat travels 2.4 miles downstream
and in the same amount of time, it travels 1.8 miles upstream. If the boat travels at an average
speed of 21 miles per hour in the still water, find the current of the river.
(SHOW WORK)
Current x miles per hr
Downstream speed (21+x)
t= 2.4
21+ x
upstearem speed (21−x)t= 1.8
21−x
Since time is equal 2.4
21+ x = 1.8
21−x
50.4−2.4 x=37.8+1.8 x
12.6=4.2 x
x=3
The current of the river is 3 miles per hour
Sandra swims the 100-meter freestyle for her school’s swim team. Her state’s ranking system
awards 3 points for first place, 2 points for second, 1 point for third, and 0 points if she does not
place. Her coach used her statistics from last season to design a simulation using a random
number generator to predict how many points she would receive in her first race this season.
Integer Value Points Awarded Frequency
1 - 8 3 20
9 - 15 2 12
16 - 19 1 6
20 0 2

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
9. What is Sandra’s expected value of points awarded for a race?
(SHOW WORK)
Expected value of points=∑ fx
f = 90
40 =2.25
A carnival game has the possibility of scoring 50 points, 75 points, or 150 points per turn. The
probability of scoring 50 points is 60%, 75 points is 30%, and 150 points is 10%. The game
operator designed a simulation using a random number generator to predict how many points
would be earned for a turn.
Integer Value Points Frequency (f) Prob fx
0 - 5 50 55 0.6 1650
6 - 8 75 32 0.3 720
9 150 13 0.1- 195
Total 100 2565
10. What is game’s expected value of points earned for a turn?
(SHOW WORK)
Expected points
∑ fx
f =2565
43.9 =58.43

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
11. A research company wants to test the claim that a new multivitamin helps to improve short
term memory. State the objective of the experiment, suggest a population, determine the
experimental and control groups, and describe a sample procedure.
(SHOW WORK)
Objective: To ascertain the claim that a new multivitamin helps to improve short term memory.
Population: The entire group of people with memory loss sickness.
Experimental group: the patients who did not receive the multivitamin.
Control group: The patients who are put on the multivitamin subscription.
Sample procedure; Identify a hospital with memory loss patients and sample the patients from
the hospital.
A medical team sent surveys to randomly selected households to determine the various health
problems. The result of the survey is shown below.
Health Problems Number of Patients
Obesity 32
Diabetes 54
Heart problems 78
Eye problems 112
Dental problems 96
Total 372
Note: The survey result for each health problem is mutually exclusive.
12. Based on the survey, what is the probability that a person chosen at random is a diabetic
patient or an eye patient?

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
(SHOW WORK)
Pr ( diabetic )= 54
372= 9
62
Pro ( eye patient )= 112
372 = 28
93
probability that a person chosen at random is a diabetic patient or an eye patient
¿ 9
62 + 28
93 = 83
186 =0.4462
For a short time after a wave is created by wind, the height of the wave can be modeled using y
= a sin , where a is the amplitude and T is the period of the wave in seconds.
13. Write an equation for the given function given the amplitude, period, phase shift, and vertical
shift.
amplitude: 4, period 4 phase shift = vertical shift = -2
The equationwill be
y=4 sin ( 2 πt
4 π − 4
3 π ) −2
y=4 sin ( 1
2 t− 4
3 π )−2

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
14. Write an equation for the given function given the period, phase shift, and vertical shift.
cotangent function, period phase shift vertical shift
y=sin ( 2 πt
π − 1
3 π )+2
y=sin (2 t−1
3 π )+2
15. Use the unit circle to find the value of and .
π=180 degrees
3 π
2 =270
This falls in the third quadrant of the unit circle.
Sin y
1 =−1
1 =−1
Cos ¿ A
H =−0
1 =0
16. Suppose θ is an angle in the standard position whose terminal side is in Quadrant IV and

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
. Find the exact values of the five remaining trigonometric functions of θ.
(SHOW WORK)
From the unit circle
cot ( θ ) =cos
sin = adj
opp =−6
7
sin ( θ ) = op
Hyp = 7
√85
cos ( θ )= −6
√85
tan ( θ )= sin
cos =
7
√85
−6
√85
=−7
6
Csc ( θ ) = 1
sin ( θ ) = √ 85
7
17. Verify is an identity.
First we get the square root of bot sides
which gives :sin2 x−sinx=cos2 x−cosx
sin2 x−cos2 x=sinx−cosx
cos2 x−sin2 x=cosx −sinx
cos 2 x=cosx−sinx
cos 2 x−cosx=−sinx
cosx=−sinx
This is a proof of identity

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
18. Find all solutions of each equation on the interval .
tan2 x sec2 x+2 ( tan2 x +1 )−tan2 x=2
tan2 x sec2 x+2 tan2 x +2−tan2 x=2
tan2 x sec2 x+ tan2 x =0
sec2 x +1=0
sec2 x=−1
1/( cos¿¿ 2 x)=−1 ¿
−1 cos2 x=1
cosx=1∨−1
x=0 , 180
19. Solve on the interval .
sinθ−cos 2 θ+1=0
¿ sinθ=1
θ=90
20. Verify .
sin (360−θ)=−sinθ
sin 360 cosθ−cos 360 sinθ=−sinθ

Name:
Algebra 2 Part 2 Final Exam
Directions: Answer the questions below. Make sure to show your work and justify all your
answers.
sin 360=0
0−sin θ=−sinθ
sin θ=sinθ