Pre-Calculus Midterm Examination: Foundations of Mathematics Problems

This document presents a comprehensive solution to a Pre-Calculus midterm examination. It includes detailed answers to various problems, such as solving trigonometric equations, calculating population growth, analyzing stellar magnitudes, determining angular velocity, and finding domains and ranges of functions. The solution also covers synthetic substitution, trigonometric function evaluation, and calculating the area of a triangle. Each problem is accompanied by step-by-step workings and justifications, offering a thorough understanding of the concepts and methods. The assignment covers a broad range of pre-calculus topics, demonstrating a strong grasp of mathematical principles and their applications.

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
Find the solutions of each equation on the interval [0, 2π).
1.
(SHOW WORK)
= 2 sin (
3 π
2 + x+ 3 π
2 −x
2 )cos (
3 π
2 + x− 3 π
2 + x
2 )
= -2
= 2 sin ( 3 π
2 )cos (x )= -2
= -cos(x) =-1
= cos x = 1
x=0 , 2 π
2. The following table shows the estimated populations and annual growth rates for four countries in the year 2000. Find the
expected population of each country in 2025, assuming their annual growth rates remain steady.
Country Population in 2000 Growth Rate
Australia 19,169,000 0.6%
China 1,261,832,000 0.9%
Mexico 100,350,000 1.8%
Zaire 51,965,000 3.1%

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
(SHOW WORK)
Constant growth rate follows the simple interest formula where
p = population in 2000, n= (2025 – 2000)= 25, r = growth rate
Expected population of Australia in the year 2025 : 22044350
Expected population of China in the year 2025 : 1545744200
Expected population of Mexico in the year 2025: 145507500
Expected population of Zaire in the year 2025: 92237875
3. Solve .
(SHOW WORK)
5
√249−2 x=1
= Raising to the 5th power,
249 – 2x =1
2x = 248
x = 124
Estimate and classify the critical points for the graph of the function.

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
4.
Answer:
Critical points: (-0.5 , 1, -3)
5. Astronomers classify stars according to their brightness by assigning them a stellar “magnitude.” The higher the magnitude
the dimmer the star. The dimmest stars visible to the naked eye have stellar magnitudes of 6. The table below shows the
relative brightness of different stellar magnitudes.
Stellar Magnitude 1 2 3 4 5 6
Relative Brightness 100 40 16 6.3 2.5 1
a. Find an equation that gives the relative brightness in terms of stellar magnitude.
b. Use this equation to find the relative brightness of a star with magnitude 9.
(SHOW WORK)
Here the relative brightness of the stars form a geometric progression with common ratio 0.4.
40
100 =16
40 = 6.3
16 =2.5
6.3 = 1
2.5 =0.4

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
a)
Therefore the nth term of the series and the equation that will give the relative brightness of different stellar
magnitudes is: 1*(0.4)n.
b)
Relative brightness of a star with magnitude 9: (0.4)9 = 0.00026
6. A restaurant offers a lunch special in which a customer can select from one of the 7 appetizers, one of the 10 entrees, and
one of the 6 desserts. How many different lunch specials are possible?
(SHOW WORK)
Answer:
A customer can choose from 7 appetizers in 7 ways.
A customer can choose from 10 entrees in 10 ways.
A customer can choose from 6 desserts in 6 ways.
Total number of lunch specials possible : 7*10*6 =420.

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
Divide using long division.
7.
(SHOW WORK)
−3 x5+11 x4 +33 x3−26 x2−36 x−6
¿−3 x2 ( x3+6 x2 −3 x −5 ) +29 x4 +24 x3−41 x2−36 x−6
−3 x2 ( x3 +6 x2−3 x−5 ) +29 x ( x3 +6 x2−3 x−5 ) +−150 x3 +46 x2 +109 x−6
−3 x2 ( x3 +6 x2−3 x−5 ) +29 x ( x3 +6 x2−3 x−5 ) −150 ( x3+ 6 x2−3 x−5 ) +946 x2−341 x −756
(−3 x2 +29 x −150 ) ( x3 +6 x2−3 x−5 ) +946 x2−341 x−756

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
Condense each expression.
8.
= log5 x5−log5
4
√ 8−x
= log5
4
√ (8−x) x20
9. A gear of radius 6.1 cm turns at 11 revolutions per second. What is the linear velocity of the gear in meters per second?
(SHOW WORK)
The linear velocity at a radial distance r is given by:

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
v=w × r
Where w is the angular velocity given by, w = 2πN ( N is the number of revolutions per sec)
= 2π × 6.1
= 38.33 rad/sec
Therefore, linear velocity : v = 38.33 * (6.1/100)
= 2.34 ms-1
Use the graph to determine the domain and range of the relation, and state whether the relation is a function.
10.
Domain: (0, ∞)
Range : (-∞,∞)
No, as the function has multiple values at same inputs this relation does not define a function.

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
11. Simplify .
(SHOW WORK)
Answer:
= tan (9 x−5 x)
= tan ( 4 x)
Find each f(c) using synthetic substitution.
12. f(x) =5x5 + 10x4 + 3x3 + 8x2 – 6x – 3; c = 3
(SHOW WORK)
f ( 3 )=5∗( 3 )5 +10∗34 +3∗33 +8∗32−6∗2−3
= 5* 243 + 10* 81 +3* 27 +8* 9 – 12 -3
=2163

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
13. Suppose θ is an angle in the standard position whose terminal side is in Quadrant I and . Find the exact values
of the five remaining trigonometric functions of θ.
All the trigonometric ratios in the first quadrant is positive.
cosθ= √¿ ¿
tanθ= sinθ
cosθ =84 /85
13
85
= 84
13
secθ= 1
cosθ = 85
13
cosecθ= 1
sinθ = 85
84
cotθ= 1
tanθ = 13
84
14. A truck driver travels at 59 miles per hour. The truck tires have a diameter of 30 inches. What is the angular velocity of
the wheels in revolutions per minute (rpm)?
(SHOW WORK)
Radius of the truck tire: (30/2) = 15 inches = 38.1 cm
Linear velocity of the wheel = 59 mph = 26.37 ms-1
Angular velocity of the wheel = ( v/r ) = (26.37 * 100) / 38.1 rpm = 69.21 rpm

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
15. Determine the zeros for and the end behavior of f(x) = x(x – 4)(x + 2)4.
The zeroes of the function f(x) are 0, 4, -2
16. Verify .
(SHOW WORK)
Answer:
sin ( 180−θ )=sin 180. cosθ−cos 180. sinθ =0− (−1. sinθ )=sinθ
17. The hourly temperature at Portland, Oregon, on a particular day is recorded below.
1 A.M. 2 3 4 5 6 7 8 9 10 11 12 Noon
46° 44° 43° 41° 40° 40° 41° 43° 46° 52° 65° 69°
1 P.M. 2 3 4 5 6 7 8 9 10 11 12 Midnight
72° 74° 75° 75° 77° 75° 74° 70° 62° 55° 51° 48°

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
a. Find the amplitude of a sinusoidal function that models this temperature variation.
b. Find the vertical shift of a sinusoidal function that models this temperature variation.
c. What is the period of a sinusoidal function that models this temperature variation?
d. Use t = 0 at 5 P.M. to write a sinusoidal function that models this temp. variation.
e. What is the model’s temperature at 10 A.M.? Compare this to the actual value?
Amplitude : (77-40)/2 = 18.5
Vertical Shift: (77+40)/2 = 58.5
2 π
b =24 , b = π
12
Phase = π /12
Find intersections and unions of the following given sets.
18. Ten students from a school appear in one or more subjects for an inter school quiz competition as shown in the table
given below.
General
Knowledge Math Science
Acel Barek Carlin
Acton Bay Acton
Anael Max Anael
Max Kai Kai
Carl Anael Dario
Dario Carlin Barek
Let G represents the set of students appearing for General Knowledge, M represents the set of students appearing for Math,
and S represents the set of students appearing for Science.
Find and .
= ( Anael) , = (Acel, Acton,Anael, Max,Carl, Dario, Carlin, Kai, Barek)

Name:
Pre-Calculus Midterm
Directions: Answer the questions below. Make sure to show your work and justify all your answers.
19. Heather invests $4,900 in an account with a 3.5% interest rate, making no other deposits or withdrawals. What will
Heather’s account balance be after 5 years if the interest is compounded 2 times each year?
(SHOW WORK)
A=P (1+ r
n )nt
=4900 (1+ 0035
2 )2× 5
=5,828.28
A
ccount balance be after 5 years : $5828.28
20. Find the area of the triangle with a = 19, b = 14, c = 19. Round to the nearest tenth.
(SHOW WORK)
The triangle is isoceles as two of its sides are equal. Considering the base to be 14 the length of the perpendicular
drawn from the vertex where the equal sides meet = √ 192−¿ ¿
Therefore area of the triangle = 1
2 ×base ×altitude= 1
2 ×7 ×17.66=61.81