Republic of the Philippines
GILLESANIA Engineering Review and Training Center
Cebu
BOARD OF CIVIL ENGINEERING
MATHEMATICS, SURVEYING & TRANSPO. ENG’G.
Wednesday, November 29, 2017
SOLUTIONS
1.
SET B
Module 8 Solutions
A 17-foot ladder is sliding down a wall at a rate of -5 feet/sec.
When the top of the ladder is 8 feet from the ground, how fast is
the foot of the ladder sliding away from the wall (in feet/sec)?
A. 7/8
C. 5/3
B. 8/3
D. 17/5
1
F=

rt
A e 1

r
e 1
Given
12.61 e
500 =
i( 39)
 1
i  Find( i)  0.00087
i
e 1
i  0.087 %
m 
365
y
7
 52.14
weeks
rc  ( 52 ) i  4.52 %
x
4.
𝑥 = √𝐿2 − 𝑦 2
𝑑𝑥 𝑑𝑥 𝑑𝑦
=
𝑑𝑡 𝑑𝑦 𝑑𝑡
2.
A 145-g ball is thrown so that it acquires a speed of 25 m/s.
What was the net work done on the ball to make it reach this
speed, if it started from rest?
A. 36 J
C. 54 J
B. 45 J
D. 63 J
4
𝑑𝑥
8
= -0.5333 =
𝑑𝑦
15
S ince the initial kinetic energy was zero, the net
work done is equal to the final kinetic energy.
𝑑𝑥
8
8
= - × (−5) = 𝑓𝑒𝑒𝑡/𝑠𝑒𝑐
𝑑𝑡
15
3
KE =
A television game show has three payoffs with the following
probabilities:
Payoff ($)
0
1000
10,000
Probability
.6
.3
.1
What are the mean and standard deviation for the payoff
variable?
Hint: 𝜇𝑥 = σ 𝑥𝑓(𝑥) and σx² = σ 𝑥 2 𝑓(𝑥) − 𝜇𝑥 2
A. μx = 1300, σx = 2934
B. μx = 1300, σx = 8802
2
1
KE 
5.
C. μx = 3667, σx = 4497
D. μx = 3667, σx = 5508
mv
2
1
2
2


( 0.145 kg )  25
m
2
  45.31 J
s
A block weighing W = 500 lb rests on a ramp inclined 29° with
the horizontal. What minimum force must be applied to keep
the block from sliding down the ramp? Neglect friction.
A. 422 lb up the ramp
C. 242 lb up the ramp
B. 347 lb up the ramp
D. 437 lb up the ramp
5
F  500 sin( 29°)  242.4
μx = 0(.6) + 1000(.3) + 10,000(.1)
μx = 1,300
σ² = σ 𝑥 2 𝑓(𝑥) − 𝜇2
σ² = 0²×(.6) + 1000²×(.3) + 10,000²×(.1) - 1,300²
σ² = 8,610,000
σ = 2,934.3
3.
A bank offers its customers a Christmas Club account, in which
they deposit $12.61 a week for 39 weeks, starting in midFebruary. At the end of 39 weeks (mid-November), each
customer will have accumulated $500, which can be withdrawn
to pay for gifts and other seasonal expenses. What is the
nominal interest rate, assuming continuous compounding?
A. 6.12%
C. 5.32%
B. 5.78%
D. 4.52%
3
6.
Find the upper base of an isosceles trapezoid if the area is 52ξ3
the altitude has length 4ξ3, and each leg has length 8.
A. 9
C. 11
B. 10
D. 12
6
A =
a b
x 
2
8
2
52 3 =
a 9
a
h
8
x
2
  4 3  4
 a  ( a  2  4)   4


2


4ξ3
b = a + 2x
3
7.
On its first pass, a pendulum swings through an arc whose
length is 24 inches. On each pass thereafter, the arc length is
75% of the arc length on the preceding pass. Find the total
distance the pendulum travels before it comes to rest.
A. 90 inches
C. 94 inches
B. 92 inches
D. 96 inches
7
𝑎1
𝑆=
1−𝑟
24
𝑆=
1 − 0.75
𝑆 = 96 inches
11. If the probability of a spacecraft being struck by exactly one
cosmic particle during and Earth-Neptune roundtrip is identical
to its probability of not being struck at all, what is this
probability?
A. 0.351
C. 0.368
B. 0.531
D. 0.638
 11
Assuming a Poisson distribution of collisions,
the probability that exactly x collision is:
P=
e
8.
Three circles with radii 3.0, 5.0, and 9.0 cm are externally
tangent. What is the area of the triangle formed by joining their
centers?
A. 45 cm²
C. 48 cm²
B. 56 cm²
D. 52 cm²
8
a  358
b  3  9  12
c  5  9  14
s
8  12  14
2
 17
A 
17 ( 17  8 ) ( 17  12 ) ( 17  14 )
9.
The length of a rectangular playing field is 5 meters less than
twice its width. If 230 meters of fencing enclose the field, what
are its dimensions?
A. 40 m by 75 m
C. 45 m by 70 m
B. 48 m by 67 m
D. 38 m by 77 m
9

x

=
x
e
 0

0
=1
P  e
1
 0.368
12. Mr. Holzman estimates that the maintenance cost of a new car
will be $75 the first year, and will increase by $50 each
subsequent year. He plans to keep the car for 6 years. He wants
to know how much money to deposit in a bank account at the
time he purchases the car, in order to cover these maintenance
costs. His bank pays 5½% per year, compounded annually, on
savings deposits.
A. $985.17
C. $979.08
B. $969.56
D. $960.17
 12
6
P 

x
A  47.91
 x
 x
s( s  a) ( s  b ) ( s  c )
A =
e
25  50x
1
1.055
x
 960.17
13. Southern Star Realty is an established real estate company that
has enjoyed constant growth in sales since 1995. In 1997 the
company sold 200 houses, and in 2002 the company sold 275
houses. Use these figures to predict the number of houses this
company will sell in the year 2011.
A. 400 houses
C. 420 houses
B. 410 houses
D. 430 houses
 13
Given
x = 2y  5
2x  2y = 230
Find( x y) 
 75 
 
 40 
10. A conical drinking cup has a 12-inch rim and is 4 inches at the
center. If creased flat, what is the vertex angle of the resulting
figure?
A. 66° 55’
C. 88° 22’
B. 77° 33’
D. 55° 44’
 10
C  12in
h  4in
C = 2 r
r
R
 
C
2
 1.91 in
2
2
r  h  4.43 in
0.5 C
R
 77.56 deg
 77 
 33  DMS



 24.26 
14. Find the length of a diagonal of a rhombus if the other diagonal
has length 8 and the area of the rhombus is 52.
A. 12
C. 14
B. 13
D. 15
 14
A = ½ d₁ d₂
52 = ½ d₁ (8)
d₁ = 13
15. A person can choose between two charges on a checking
account. The first method involves a fixed cost of $11 per
month plus 6¢ for each check written. The second method
involves a fixed cost of $4 per month plus 20¢ for each check
written. How many checks should be written to make the first
method a better deal?
A. more than 50 checks
C. more than 60 checks
B. less than 50 checks
D. less than 60 checks
 15
$11 + 0.06x < $4 + 0.2x
$11 - $4 < 0.2x – 0.06x
$7 < 0.14x
X > 50 (more than 50 checks)
16. A corner lot of land is 122.5 m. on one street and 150 m. on the
other street, the angle between the two streets being 75°. The
other two lines of the lot are respectively perpendicular to the
lines of the streets. What is the perimeter of the boundary of
the lot?
A. 452.22 m
C. 307.16 m
B. 372.50 m
D. 481.60 m
 16
a
122.5
cos ( 75°)
19. Of the coral reef species on the Great Barrier Reef off Australia,
73% are poisonous. If at mist boat taking divers to different
points off the reef encounters an average of 25 coral reef
species, what are the mean and standard deviation for he
expected number of poisonous species seen?
A. μx = 6.75, σx = 4.93
C. μx = 18.25, σx = 4.93
B. μx = 18.25, σx = 2.22
D. μx = 18.25, σx = 8.88
 19
p  73%  0.73
 150  323.3
b  atan ( 15°)  86.63
c  122.5 tan ( 75°) 
c
90°
a
n  25
b
75°
150 m
15°
a
cos ( 15°)
x  n p  18.25
x 
P  150  122.5  b  c  481.6
17. What is the perimeter of r = 3(1 + cos θ)?
A. 6π
C. 18
B. 13.5π
D. 24
 17
PP =
r = 3  3 cos ( )
d
n p q  2.22
20. A proposed manufacturing plant will require a fixed capital
investment of P8,000,000 and an estimated working capital of
P1,500,000 M. The annual profit is P2,000,000 and the annual
depreciation is to be 8% of the fixed capital investment.
Compute the payout period.
A. 2.89 years
C. 3.67 years
B. 3.03 years
D. 4.13 years
 20
c  122.47
dr
q  1  p  0.27
PP 
=  3 sin( )
FC
AP  AD
8000000
2000000  0.08 ( 8000000 )
PP  3.03

L  

2
2
2
( 3  3 cos ( ) )  (  3 sin( ) ) d 
0
L  24
18. The rate of change of the temperature of an object is
proportional to the difference between the object’s
temperature and the temperature of the surrounding medium.
Assume that a refrigerator is maintained at a constant
temperature of 45 °F and that an object having a temperature
of 80 °F is placed inside the refrigerator. If the temperature of
the object drops from 80 °F to 70°F in 15 minutes, how long
will it take for the object’s temperature to decrease to 60 °F?
A. It would take 22.7727 min for the its temperature to drop
from 70° to 60°.
B. It would take 27.7727 min for the its temperature to drop
from 70° to 60°.
C. It would take 32.7727 min for the its temperature to drop
from 70° to 60°.
D. It would take 37 7727 min for the its temperature to drop
from 70° to 60°.
 18
When x = 0, y = 80 – 45 = 34
When x = 15, y = 70 – 45 = 25
21. The observed interior angles of a quadrilateral and their
corresponding number of observation are as follows:
NO. OF
CORNER
ANGLE
OBSERVATIONS
1
67°
5
2
132°
6
3
96°
3
4
68°
4
Determine the corrected angle at corner 3.
A. 95°37.86’
C. 95°52.96’
B. 94°56.84’
D. 94°12.55’
 21
A  67° B  132 °
D  68°
  A  B  C  D  360 °  3 °
The sum exceeds 360deg, then the correction must be
structed from the measured angles.
1
c  3 °
3
1
5

1
6

1
3

1
 1.05 °
4
Ccorr  C  c  94.95 °
Solve for x when y = 60 – 45 = 15
Ccorr
t = 37.77 – 15 = 22.77
C  96°
 94 
 56  DMS



 50.53 
22. An engineering firm has turned to Friendly Shark, Inc., to borrow
$30000 needed for a short-term (2-year) project, attracted by an
advertisement announcing an interest rate of 12% per year. Friendly
Shark's loan statement indicates the following:
Interest: ($30 000) (1% per month)(24
$ 7 200
months)
Loan
30 000
Total
$37 200
Monthly installment = $37 200/24 =
$1550
What is the actual cost of borrowing money from Friendly Shark, Inc.?
A. 23.87%
B. 21.57 %
C. 22.67%
D. 24.83%
 22
P=


i( 1  i)
 A in  F in
 

G
  A

F = A 1  G
n
A ( 1  i)  1
F  ( 1000  200  7.2453 ) ( 42.1359 )  103193.35
n
Given
1550  1  i
30000 =
i 1  i
24
20
 1
P 
24
x
i  0.018
12
 1  0.2387
re  23.8721 %
23. The cost of fuel to run a locomotive is proportional to the
square of the speed and $25 per hour for a speed of 25 miles
per hour. Other costs amount to $100 per hour, regardless of
the speed. Find the speed that minimizes the cost per mile.
A. 20 mi/h
C. 40 mi/h
B. 30 mi/h
D. 50 mi/h
 23
2
25
25
c t = 100  0.04 v
2
 0.04
2

d
dv
S
RP 
2
v
2
CT( v)  0.08 
0.04 v  100
v
2
2
0.04 v  100
  50.0 


 50.0 
v
2
=0
8.5
2.8  1.85
RP  8.95
27. Solve for real values of x:
𝑥
𝑥
൫7 + 4ξ3൯ − 4൫2 + ξ3൯ = -1
A. 0
B. 1
C. 1, -1
D. -1, 0
 27
3 = 1
x
x  1.0
25. Ms. Brown deposits Php1000 in the bank at the end of the first
year, Php1200 at the end of the second year, etc., continuing to
increase the amount by Php200 a year, for 20 years. If the bank
pays 7% per year, compounded continuously, how much
money will have accumulated at the end of 20 years?
A. Php105,823.81
B. Php100,982.63
 25
FC
NAP
 7  4 3 x  4  2 
24. The sum of two numbers is 48, and the sum of their reciprocals
is 16. Find their product.
A. 3
B. 4
C. 5
D. 6
 24
𝑥+𝑦 𝑥+𝑦
=
= 𝑥𝑦
1 1 𝑦+𝑥
+
𝑥𝑦
𝑥 𝑦
48
𝑥𝑦 =
=3
16
Use the following factors:
Future worth factor:
Gradient Uniform Series
 103193.69
TAE  1.85
dC( v) = 0
0.08 
0.07 20
TAE  1  0.6  0.2  0.05
S  1
CT( v)  100  0.04 v
v 
1
 25447.25
0.07x
26. An investment of. P8.5 M is expected to yield an annual income
of P2.8 M. Determine the recovery period in years based on the
following estimates.
Annual depreciation = P1.0 M
Operational expenses = P0.6 M
Taxes and insurance = P0.2 M
Miscellaneous expenses = P50,000
A. 8.5 years
C. 8.9 years
B. 8.7 years
D. 9.1 years
 26
RP =
k 
dC( v) 
e
F  25447.25 e
re  ( 1  i)
c f = kv
cf
k=
2
v
800  200 x

[F/A, 7%, 20] = 42.1359
[A/G, 7%, 20] = 7.2453
C. Php103,193.35
D. Php111,620.88
28. The #XanderFord fans club decides to play the game of craps. A
pair of dice is rolled in this game and the sum to appear on the
dice is of interest. What is the mathematical expectation of the
sum to appear when the dice are rolled?
A. 6
C. 8
B. 7
D. 9
 28
EXP  2
1
36
3
2
36
4
3
36
5
4
36
6
5
36
7
6
36
8
5
36
9
4
36
 10
3
36
 11
2
36
 12
1
36
EXP  7
29. A boat is being pulled toward a dock by a rope attached to its
bow through a pulley on the dock 7 feet above the bow. If the
rope is hauled in at a rate of 4 ft/sec, how fast is the boat
approaching the dock when 25 ft of rope is out?
A. -25/3 ft/s
C. -25/6 ft/s
B. -25/4 ft/s
D. -25/7 ft/s
 29
𝑥 2 = 𝐿2 − 𝑦 2
𝑑𝑥
𝑑𝐿
2𝑥
= 2𝐿
−0
𝑑𝑡
𝑑𝑡
𝑑𝑥
2 √252 − 72 ( ) = −2(25)(-4)
𝑑𝑡
𝑑𝑥
200
25
==𝑑𝑡
2(24)
6
30. What is the least number of persons required if the probability
exceeds ½ that two or more of them have the same birthday?
(Year of birth need not match.)
A. 18
C. 28
B. 23
D. 32
 30
What is the least number of persons required if the
probability exceeds ½ that two or more of them have the
same birthday? Answer: 23
35. It is advisable for a site plan to contain a large scale map of the
overall area and to indicate where the project is located on the
site.
A. Location Map
C. Vicinity Map
B. Site Plan Map
D. Google Map
 35
31. Calculate the impulse experience when a 70-kg person lands on
firm ground after jumping from a height of 3.0 m.
A. 570 N∙s
C. 550 N∙s
B. 560 N∙s
D. 540 N∙s
 31
36. A rectangle is to be inscribed in a semicircle with radius 4, with
one side on the semicircle’s diameter. What is the largest area
this rectangle can have?
A. 16
C. 24
B. 21
D. 25
 36
Impulse = Momentum
v=
2g h
v 
2 ( 9.81 ) ( 3 )  7.672
Momentum  70  7.672  537.04 J
32. From the top of a lighthouse, 175 ft above the water, the angle
of depression of a boat due south is 18°50’. Calculate the speed
of the boat if, after it moves due west for 2 min, the angle of
depression is 14°20’.
A. 234 ft/min
C. 227 ft/min
B. 222 ft/min
D. 244 ft/min
 32
let b = 2x (the base of the rectangle)
y = height
y=
4 x
2
A = 2xy
2
dA ( x) 
175 ft
d
dx
2
2
A ( x)  2 16  x 
2x
2
2 16  x 
AC 
BC 
v 
175 ft
tan ( DMS ( 14 20 ) )
2
2
2
2
16  x
tan ( DMS ( 18 50 ) )
2x
16  x
Given
175 ft
y
2x
2
A ( x)  2x 4  x
AB 
(x, y)
2
=0
x  Find( x)  2.83
 513.08 ft
x1 8
 684.887 ft
y 
2
2
4 x 1 8
A  2xy  16
2
( maximum)
AC  AB  453.673 ft
BC
2min
 226.836
ft
min
33. Solve for x: log x² = 1 + (log x)².
A. 1/10
B. 1
C. 10
 33
Let y = log x
log x² = 2 log x = 2y
D. 100
2y = 1 + y²
y² - 2y + 1 = 0
(y – 1)² = 0
y=1
log x = 1, then x = 101 = 10
34. An engineer selects a sample of 5 iPods from a shipment of 100
that contains 5 defectives. Find the probability that the sample
contains at least one defective.
A. .230
B. .211
C. .286
D. .271
 34
C( x y)  combin( x y)
C( 95 5)
P  1
 0.23
C( 100 5)
37. The offset distance from PC to PT of a simple curve is 18 m and
the angle of intersection of the tangents is 24°. If the stationing
of the PT is 45 + 158.32, what is the stationing of the PI?
A. 45 + 123.44
C. 45 + 115.30
B. 45 + 109.78
D. 45 + 106.28
 37
18 = R(1 – cos 24°)
R = 208.2019002031 m
Lc = πRI/180°
Lc = 87.211408018866
T = R tan I/2
T = 44.25 m
STA PI = STA PT – Lc + T
STA PI = 45+115.30
38. A cylindrical tank with a radius of 6 meters is filling with fluid
at a rate of 108π m³/sec. How fast is the height increasing?
A. 2π m/s
C. 2 m/s
B. 3π m/s
D. 3 m/s
 38
dh/dt = v = Q/A
dh/dt = 108π/π(6²) = 3 m/s
39. As reported in The New York Times (February 19, 1995, p.-12),
the Russian Health Ministry announced that one-quarter of the
country's hospitals had no sewage s stem and one-seventh had
no running water. What is the probability that a Russian
hospital will have at least one of these problems
a. If the two problems are independent?
b. If hospitals with a running water problem are a subset
of those with a sewage problem?
A. 11/28, 1/7
C. 9/28, 1/7
B. 91/28, 1/4
D. 5/14, 1/4
 39
1
1
Ps 
4
7
P  P s 1  P w    1  P s P w  P s P w
Pw 
P
40. This calculates an adjustment that is applied to each latitude
and each departure individually. A proportion is established
that uses the length of the line, the perimeter, and the closure.
In general form, the formula is:
𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
=
𝐶𝑙𝑜𝑠𝑢𝑟𝑒
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟
A. Transit Rule
C. Closure Rule
B. Compass Rule
D. Closed Traverse Golden Rule
 40
Compass Rule - Calculates an adjustment that is applied
to each latitude and each departure individually. A
proportion is established that uses the length of the line,
the perimeter, and the closure.
41. The areas of two similar polygons are 80 and 5. If a side of the
smaller polygon has length 2, find the length of the
corresponding side of the larger polygon.
A. 24
C. 32
B. 16
D. 8
 41
 s
 
 2
2
80
=
s 2
43. A ball is thrown vertically upward from the top of the Leaning
Tower of Pisa (190 feet high) with an initial velocity of 96 feet
per second. The function
s(t) = -16t² + 96t + 190
models the ball’s height above the ground, s(t), in feet, t
seconds after it was thrown. During which time period will the
ball’s height exceed that of the tower?
A. Between 0 and 5 seconds C. Between 0 and 7.6 seconds
B. Between 0 and 6 seconds D. Between 0 and 8.2 seconds
 43
2
s( t )   16 t  96t  190
300
s( t)
200
100
0
0
2
4
6
8
10
t
2
 16 t  96t  190  190
0t 6
5
80
5
8
42. Calculate the power required of a 1400-kg car under the
following circumstances:
(a) the car climbs a 10° hill (a fairly steep hill) at a steady
80km/h; and
(b) the car accelerates along a level road from 90 to 110
km/h in 6.0 s to pass another car.
Assume the average retarding force on the car is FR = 700 N
throughout.
A. 97 hp, 85 hp
C. 91 hp, 82 hp
B. 85 hp, 97 hp
D. 82 hp, 91 hp
 42
m
M  1400 kg
g  9.81
  10°
v  80kph
2
s
W  Mg  13.73 kN
FR  700 N
44. Cebu Pacific plane flew from Busan, Korea, whose latitude is
14°N and longitude of 121°30'E on a course S30°W. and
maintaining a uniform altitude. At what longitude will it cross
the equator?
A. 111°11’E
C. 113°33’E
B. 112°22’E
D. 114°44’E
 44
B
30°
c
C
A
b
F  W sin( )  FR  3084.88 N
P  Fv  68552.98 W
P  91.93 hp
m
vi  90kph  25
s
t  6s
a
vf  vi
t
 0.93
vf  110 kph  30.56
m
2
s
F  Ma  FR  1996.3 N
P  Fvf  60997.94 W
P  81.8 hp
m
s
a = 14°
θ
Using Napier’s Rule:
sin a = tan Bc tan b
sin 14° = tan b/tan 30°
b = 7°57’
θ = 121°30’ - 7°57’
θ = 113°33’E
Bc
cc
Ac
a
b
12x
f ( x) 
2
x 4
4
3
f( x)
2
1
0
1
0
1
2
3
4
5
6
x
4
 12x
A  
d x  8.318
 x2  4

6 ln( 4 )  8.318
1
47. On the East Coast, it is known from health records that the
probability of selecting an adult over 40 years of age with
cancer is 0.05. The probability of diagnosing a person with
cancer as having the disease is 0.78 and the probability of
incorrectly diagnosing a person without cancer as having the
disease is 0.06. Find the probability that a person is diagnosed
as having cancer.
A. 0.57395
C. 0.069
B. 0.59375
D. 0.096
 47
P(D) = .05(.78) + .95(.06) = .096
48. Ship A is sailing due south at 16 mi/h, and ship B, 32 miles
south of A, is sailing due east at 12 mi/h.
(a) At what rate are they approaching or separating at the
end of 1 hour?
(b) At the end of 2 hours?
A. (a) They are approaching at 5.6 mi/h. (b) They are
separating at 12 mi/h.
B. (a) They are approaching at 5.6 mi/h. (b) They are
separating at 15 mi/h.
C. (a) They are approaching at 6.5 mi/h. (b) They are
separating at 12 mi/h.
D. (a) They are approaching at 6.5 mi/h. (b) They are
separating at 12 mi/h.
 48
4

12x

d x  6 ln( 4 )
 x2  4
1
46. Billy weighs 5 pounds more than Bobby and when they seesaw, Billy has to sit 1 foot closer to the center in order to
balance. When the twins, Tammy and Tommy, who weigh 35
pounds each, get on with them, Billy and Tammy sit only 6
inches closer to the center in order to balance Bobby and
Tommy. How long is the see-saw?
A. 10 feet
B. 12 feet
C. 14 feet
D. 16 feet
 46
X+5
1’
X
L/2 - 1’
L/2
B
A
X + 40
X + 35
L/2 – ½’
½’
CA
let x = weight of billy
x + 5 = weight of bobby
 L  1  = x L 
  
2
  2
( x  5) 
5L
2
5
 L  = ( 35  x  5 )  L  1 



 2
 2 2
( 35  x) 
x = 5L  40
5L
2
 5 = 5L  40
2
( 12t )  ( 32  16t )
2 ( 50t  64 )
d
v( t ) 
S( t ) 
2
dt
( 16t  32 )
2
 9t
16
v( 1 )   5.6
v( 2 )  12
16t
12t
49. Given the sides of a triangle ABC, a = 36.3 cm, b = 23.9 cm and
∠A = 77.3°. Compute the length of side c.
A. 33.08 cm
C. 42.74 cm
B. 39.96 cm
D. 49.50 cm
 49
a  36.3
b
sin( B)
=
b  23.9
A  77.3 °
a
sin( A )
 b sin( A )   39.96 °
B  asin

 a 
C    A  B  62.74 °
L/2
BC
x=
2
S( t ) 
32 - 16t
45. Find the area of the region bounded by the curves
12𝑥
𝑦= 2
𝑥 +4
the x-axis, x = 1, and x = 4.
A. 4 ln 5
C. 6 ln 4
B. 4 ln 6
D. 6 ln 5
 45
c 
a
sin( A )
sin( C)  33.08
50. The horsepower that can be safely transmitted to a shaft varies
jointly as the shaft’s angular speed of rotation (in revolutions
per minute) and the cube of its diameter. A 2-inch shaft making
120 revolutions per minute safely transmits 40 horsepower.
Find how much horsepower can be safely transmitted by a 3inch shaft making 80 revolutions per minute.
A. 80 horsepower
C. 100 horsepower
B. 90 horsepower
D. 120 horsepower
 50
P = kd
k=
L  14
k 
3
P
d
3
40
3

1
24
120  2
1
3
P 
( 80 ) 3  90
24
 
51. A record enthusiast decided to calibrate his 33⅓ rpm player by
placing equally spaced dots around the rim. What is the
minimum number of dots required in order that they appear
stationary under 60 cycle light?
A. 184
C. 216
B. 148
D. 261
 51
55. Three spheres of lead with radii r, 2r and 4r, respectively, are
melted to form a new sphere of radius R. The ratio of the
volume to the surface area of the new sphere is equal to 4.18.
Compute the radius r.
A. 2
C. 4
B. 3
D. 5
 55
V ( r) 
4
3
3
3
3
 r  ( 2r)  ( 4r) 
V ( r) 
52. How much money must initially be deposited in a savings
account paying 5% per year, compounded annually, to provide
for ten annual withdrawals that start at Php6000 and decrease
by Php500 each year?
Present worth factor:
(P/A, 5%, 10) = 7.7217
Gradient-Uniform series factor: (A/G, 5%, 10) = 4.0991
A. Php 31 426.49
B. Php30 504.19
 52
C. Php28 726.49
D. Php27 029.39
P
 A
 P

P = A  in  G in  in
A
 G
 A

6000 ( 7.7217 )  500  7.7217  4.0991  30504.19
10
P 
6500  500 x

x
1.05
1
x
P  30504.386
53. A lot has a frontage of 120 m long along a road. The other sides
which are both perpendicular to the road are 90 m and 60 m,
respectively. It is desired to subdivide the lot into two parts by
another perpendicular line to the road such that the area of the
lot that adjoins the 90-m side is equal to 1/3 of the whole area.
Determine the length of the dividing line.
A. 81.24 m
C. 83.66 m
B. 85.29 m
D. 86.89 m
 53
2
x=
n a  mb
2
n m
2
x 
1 ( 60 )  2 ( 90 )
2
12
x  81.24
54. Steve Deitmer takes 1½ times as long to go 72 miles upstream
in his boat as he does to return. If the boat cruises at 30 mph in
still water, what is the speed of the current?
A. 5 mph
C. 7 mph
B. 6 mph
D. 8 mph
 54
Let x = speed of the current
𝑆
[𝑡 = ]
𝑣
72
1
72
=1 (
)
30 − 𝑥
2 30 + 𝑥
45 − 1.5𝑥 = 30 + 𝑥
𝑥 = 6 mph
4
3
3
R =
R=
3
3
292  r
3
73 r
292 r
3
V_A ( r) =
4
292 r
4

 3 73 r 2
3
3
3
3

73 r
2
= 4.18
r  3.000474335956751449
56. Determine the speed of the earth (in mi/s) in its course around
the sun. Assume the earth’s orbit to be a circle of radius
93,000,000 mi and 1 year = 365 days.
A. 9.8 mi/s
C. 16.2 mi/s
B. 14.5 mi/s
D. 18.5 mi/s
 56
v=
v 
C
t
2  ( 93000000 mi)
365 day 
24hr
1day

3600 s
 18.529
mi
s
1hr
57. A drawer contains red socks and black socks. When two socks
are drawn at random, the probability that both are red is ½.
How small can the number of socks in the drawer be?
A. 2
C. 4
B. 3
D. 5
 57
A drawer contains red socks and black socks. When two
socks are drawn at random, the probability that both are
red is ½. How small can the number of socks in the
drawer be? 4
58. Cebu Pacific airplane travels in a direction of N30°W at an air
speed of 600 kph. If the wind has a speed of 80 kph on a
direction of N40°E, what is the ground speed of the plane?
A. 683.51 kph
C. 638.15 kph
B. 685.31 kph
D. 631.85 kph
 58
40°
vg 
2
2
600  80  2( 80) 600 cos ( 110 °)
vg  631.85
30°
vg = |600 – 80∠110°|
vg = 631.85
30°
59. There are two barrels, one containing 40 gal. of wine and 60
gal. of water, the other containing 70 gal. of wine and 30 gal. of
water. A pailful is taken from the first barrel and poured into
the second. After mixing, a pailful is poured back into the first
barrel. The proportions of win to water in the first barrel are
now 19:26. What is the capacity of the pail?
A. 5 gal
C. 7 gal
B. 6 gal
D. 8 gal
 59
 70  0.40 x  x


 100  x  = 19
26
 30  0.60 x  x
60  0.6 x  

 100  x 
40  0.40 x 
x  8.0
60. The cost of equipment is P500,000 and the cost of installation
labor, taxes and miscellaneous expenses is P30,000. If the
salvage value is 10% of the cost of equipment at the end of its
life of 5 years, compute the book value at the end of 3rd year
using MACRS Method.
A. P122,640
C. P242,000
B. P128,556
D. P146,000
 60
1 2
D3  d1  d2  d3  377360
BV 3  500000  D3  122640
61. How many people would you expect to meet before you met
one who was born on a Wednesday?
A. 6
C. 10
B. 7
D. 14
 61
Each person has a 1/7 probability of having been born on
a Wednesday. In a sense then, each person is 1/7 of an
expected Wednesday child.” Since it requires 7 such to
add up to a Wednesday child, the answer is 7 people.
62. Only two polygons can have a smallest interior angle of 120°
with each successive angle 5° greater than its predecessor. One
is the nonagon. What is the other?
A. dodecagon
C. hexadecagon
B. tetradecagon
D. octadecagon
 62
n 
 16 
 
9 
n
2
 0.3
L = 2d
L  2 ( 44.7 )  89.4 m
64. Find the perimeter of an ellipse whose second eccentricity is
0.75 and distance between foci is 6 units.
A. 25.906 units
C. 17.784 units
B. 28.448 units
D. 14.877 units
 64
2c = 6
6
c 
3
2
c
= e2
b
c
3
b=
b 
4
e2
0.75
2
a
2
3 4 5
2
a b
P  2
2
65. A traffic engineer knows that at a certain intersection over a
24-hour period, no accidents occur within probability 0.25, one
accident occurs with probability 0.60, and two or more
accidents occur with probability of 0.15. What is the probability
that over ten 24-hour periods, no accidents occur 3 times, one
accident occurs 6 times and two or more accidents occur once?
A. 0.01968
C. 0.01094
B. 0.25000
D. 0.09185
 65
3
10
6
P  0.25  0.60  0.15
361
P  0.09185
66. A bridge across a river is in the form of an arc of a circle. A boy
walking across the bridge finds that 27 feet from the shore the
bridge is 9 feet above the water. He continues on to the center
of the span and finds that the bridge is now 10 feet above the
water. How is the river?
A. 40 ft
C. 80 ft
B. 60 ft
D. 100 ft
 66
27’
9’
2
L = 2d
 28.448
2
[ 2  120  ( n  1 ) ( 5 ) ]
63. The amplitude of a deep-water wave is 2.4 m. If the depth of
water from the bottom up to the crest of the wave is 46.2 m,
determine the horizontal distance between the crests of the
wave. Assume the center rotation of the wave is 0.3 m. above
the still water level.
A. 84.90 meters C.
94.80 meters
B. 89.40 meters D.
98.40 meters
 63
46.2 d
2
R - 10
( n  2 ) 180 =
2.4
d  44.7 m

 ( 530000 )  106000
2 5

2
d2  ( 530000  106000 )  169600
5
2
d3  ( 530000  106000  169600 )  101760
5
d1 
d  46.2 
2
R = ( R  10 ) 
R=
L
L
L 
R
2
L/2
2
80
5
5=
L
R=
2
8

27L
2
2
R = ( R  1) 
2
2
80
L
 
 2
10’
L
2
8

27L
2
L

2


 27 
2
 365
 365
 80 
 
 40 
67. The average annual incomes of high school and college
graduates in a midwestern town are $21,000 and $35,000,
respectively. If a company hires only personnel with at least a
high school diploma and 20% of its employees have been
through college, what is the mean income of the company
employees?
A. $23,800
C. $28,000
B. $27,110
 67
D. $32,200
B. 2.191 mi
 71
D. 2.391 mi
EXP  21000  0.8  35000  0.2  23800
68. The clearance to an obstruction is 40 m and the desirable sight
distance when rounding a horizontal curve is 600 m. Determine
the minimum radius of horizontal curve if the length of curve is
550 m long.
A. 859.38 m
C. 1117.19 m
B. 937.5 m
D. 1218.75 m
 68
8MR = S
2
SL
8MR = L( 2S  L)
R=
R
SL
L( 2S  L)
8M
550 ( 2  600  550 )
8  40
 1117.19
69. Find the equation of the plane which makes an equal angle with
the coordinates axes and which cut a volume of 288 cubic units
from the first octant.
A. x + y + z = 12
C. x + y + z = 24
B. x +y + z = 15
D. x + y + z = 36
 69
x
2
12 = 11 ( 11  x)
23
x 
 2.091
11
12 mi
72. Which of the following are true statements?
I. The probability of an event is always at least 0 and at most
1.
II. The probability that an event will happen is always 1 minus
the probability that it won't happen.
III. If two events cannot occur simultaneously, the probability
that at least one event will occur is the sum of the respective
probabilities of the two events.
A. I and II
B. I and III
C. II and III D. I, II, and III
 72
I, II, and III
73. The length of the spiral curve is 82 m and the radius of the
central curve of the spiral curve is 260 m. Compute the length
of throw
A. 1.08 m
C. 2.16 m
B. 2.87 m
D. 4.31 m
 73
2
P=
P 
1  1 2
 a  a = 288
3 2 
a  12
x y z= k
The curve passes through P(12, 0, 0), k = 12
x  y  z = 12
70. Find the area of a piece of land with an irregular boundary as
follows:
STA
0 + 000
0 + 015
0 + 030
0 + 045
0 +060
OFFSET DISTANCE (m.)
5.59
3.38
2.30
3.96
4.80
The stations are on straight line boundary. Find the area of the
land in square meters by Simpson’s One Third Rule.
A. 225.2 m2 C. 221.75 m2
B. 227.15 m2 D. 222.5 m2
 70
d
 h1  2h odd  4h even  hn
3
15
A 
[ 5.59  2 ( 2.30 )  4 ( 3.38  3.96 )  4.8 ]
3
A=
A  221.75
71. A road is tangent to a circular lake. Along the road and 12 miles
from the point of tangency, another road opens towards the
lake. From the intersection of the two roads to the periphery of
the lake, the length of the new road is 11 miles. If the new road
will be prolonged across the lake, find the length of the bridge
to be constructed.
A. 2.091 mi
C. 2.291 mi
Ls
24R
82
2
24 ( 260 )
 1.08
74. An equipment installation job of Diego Construction in the
completion stage can be completed in 40 days of 8 hours per
day of work with 40 men working. With contract expiring in 30
days, the contractor decided to add 10 men on the job, overtime
not being permitted. If the liquidated damages are P20,000 per
day of delay and the men are paid P580 per day, compute the
total cost if he will add 10 more men to finish the job.
A. P896,000
C. P986,000
B. P869,000
D. P968,000
 74
x = no. of days to finish the ·job with 10 more men working
(40 + 10)x = 40(40)
x = 32 days
Therefore, the job is delayed by 2 days.
Penalty = 20,000(2) = P40,000
Labor cost= 580(50)(32) = P928,000
Total cost= 928,000 + 40,000 = P968,000
75. Some couples plan to hold seances around a round table.
Dropping the usual requirement that men and women
alternate, they find the number of opposite seating
arrangements is increased tenfold. How many couples are
there?
A. 3
B. 4
C. 5
D. 6
 75
let n be the number of couples
( 2n  1 )  = 10 ( n  1 ) n
n  3
( 2n  1 )   120
10 ( n  1 ) n  120