Republic of the Philippines GILLESANIA Engineering Review and Training Center
Cebu
BOARD OF CIVIL ENGINEERING MATHEMATICS, SURVEYI NG & 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 500
12.61 12.61 e
=
i( 39)
1
i F in ind( i) 0.000 0.00087 87
i
e 1 i 0.08 7 %
m
365
y
7
52.14
weeks
rc ( 52 ) i 4 .5 2 % x
4.
= √ − = = -0.5333 = 8 15 = - 8 × −5 8 / −5 = 15 3 2.
A television game show 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: and σx² =
= = σ σ
A. μx = 1300, σx = 2934 B. μx = 1300, σx = 8802 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 S ince the initial initial kinetic energy was zer o, the net work done is equal to the final kinetic energy. KE
=
KE
5.
σ −
1
mv
2
2 2
m ( 0.145kg ) 2 5 45.31 45.31 J 2 s
1
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
C. μx = 3667, σx = 4497 D. μx = 3667, σx = 5508 F 500 500 sin( 29° 29°) 2 42 .4
μx = 0(.6) + 1000(.3) + 10,000(.1) μx = 1,300 σ² =
σ −
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 At the end of 39 weeks (mid-November), each customer will have accumulated $500, which can be withdrawn to pay for gifts and other seasonal expenses.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 the altitude has length , and each leg has length 8. A. 9 C. 11 B. 10 D. 12 6
4ξ 3
A
=
a b
x 52 3 a 9
2 2
a
h 8 x
2
8 4 3 4 =
a ( a 2 4) 4 2
ξ 3
4
b = a + 2x
3
52ξ 3 52ξ
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
= 1− = 1 −240.75 75 = 96 inches
11. If the probability of a spacecraft spacecraft being struck by exactly one 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 Assumi ng a Poisso n distribution of collis ions, 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 A
8 12 14 2
17
A
17 ( 17 8 ) ( 1 7 1 2 ) ( 1 7 1 4 )
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 )
=
e
=
2 5 5 0x
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
1 2 in
C
=
r
R
4 in
2 r C
1.91 1.91 in
2 2
r
R
2
h 4.43 4.43 in
0.5C
h
77
77.56 deg deg 77.56
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 15. 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 16 122.5
a
b
a ta tan ( 1 5 °) 8 6. 6 3
c
cos( 75° )
150 323.3
122. 122.5 5 tan ( 75° 75°)
c
90°
b
75°
a
150 m
15° a
cos( 15° )
c
122. 122.4 47
P
150 122 .5 b c 481.6
=
dr d
p
73 % 0.7 3
n
25
x x
PP
=
3 3 co cos ( ) =
np
q
1 p 0.27
8. 25 25 1 8.
npq
.2 2 2 .2
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
17. What is the perimeter of r = 3(1 + cos θ)? A. 6π C. 18 B. 13.5π D. 24 17 r
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, 6.75 , σx = 4.93 C. μx = 18.25, 18.25 , σx = 4.93 B. μx = 18.25, 18.25 , σx = 2.22 D. μx = 18.25, 18.25 , σx = 8.88 19
PP
3 sin( )
FC AP AD
8000000
20000 2000000 00 0.08 0.08 ( 80000 8000000 00 )
P P 3 .0 .0 3 2
0
L
2
2
( 3 3 co cos ( ) ) ( 3 sin( ) ) d
L 24
18. The rate of change of the temperature of an object is 18. 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 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
21. The observed interior angles of a quadrilateral and their 21. corresponding number of observation are as follows: CORNER
67° 132° 96° 68° Determine the corrected angle at corner 3. A. 95°37.86’ C. 95°52.96’ B. 94°56.84’ 94°56.84 ’ D. 94°12.55’ 21 1 2 3 4
A 67°
B 13 132 °
C 96°
5 6 3 4
D 68 68 °
A B C D 360° 3 °
The sum exceeds 360deg, then the correction must be structed structed from the mea sured angles. angles. 1
c
3°
3
1 5
When x = 0, y = 80 – 45 = 34 When x = 15, y = 70 – 45 = 25
NO. OF OBSERVATIONS
ANGLE
1 6
1 3
1
1.05 ° 1.05
4
Ccorr C c 94.95 94.95 °
Solve for x when y = 60 – 45 = 15 Ccorr
94
56 DMS 50.53
22. An engineering firm has turned turned to Friendly Shark, Inc., to borrow 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
t = 37.77 – 15 = 22.77
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
n
P
=
A ( 1 i) 1 i( 1 i)
n
A 1 G A i n F i n G A
F
=
F
( 100 0 2 00 7 .2 453 ) ( 4 2.1 359 ) 103 193 .35
Given 30000
1 55 0 1 i
=
i 1 i
24
20
1
24
F 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 c f k v c f k 2 v =
2
k
25
c t 100 100 =
CT( v) dC( v)
25 2
2
0.04v 0.04v
S
0.04v
0.08
CT( v)
=
v
=
FC NAP
8.5
100
2
2.8 2.8 1.85 1.85
R P 8 .9 .9 5
27. Solve for real values of x:
൫7 + 4ξ 3൯ 3൯ −4൫2+ −4൫2+ ξ 3൯ = -1 ξ 3൯
0 2
0.08
26. An investment of. P8.5 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 2
dv
0.04v
50.0 50.0
v
100
2
=
0
B. 1
C. 1, -1
x
3 4 2
3
x =
D. -1, 0
1
x 1.0
+ = + = 1 1 + + = 48 16 = 3
25. Ms. Brown deposits Php1000 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
A. 0 27
7 4
24. The sum of two numbers is 48, 48, and the sum of their reciprocals is 16. Find their product. A. 3 B. 4 C. 5 D. 6 24
Use the following factors: Future worth factor: Gradient Uniform Series
25447.25
T AE AE 1 .8 .8 5
1
v
d
0.07x
T AE 1 0 .6 0.2 0.05
2 S 1 00 0 .0 4 v
dC( v)
v
0.04
e
2 00 00 x
0.07 0.07 20 25447 7.25 .25 e 2544 103193.69
RP
=
8 00 00
1
x
i 0 .0 1 8 re ( 1 i)
P
[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
EXP
7
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
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
= − − 0 2 = 2 2 √ 25 25 − 7 ( ) = −225-4 = - 200 = - 25 224 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 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 Impulse v
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
Momentum
=
2g h
=
v
2 ( 9 .8 1 ) ( 3 ) 7 .6 7 2
omentu um Moment
70 7.67 7.672 2 537.04 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’. 14°20’. A. 234 ft/min C. 227 ft/min B. 222 ft/min D. 244 ft/min 32
let b = 2x (the (the bas e of the rectangle) y = height y A
4 x
=
=
2
2x y 2
dA ( x) 175 ft
2
d
2
A ( x) 2 16 16 x dx
Given 2x
2
2 16 16 x
AC
BC
v
175ft
tan( DMS( 18 50 ) )
175ft
tan( DMS( 14 20 ) ) 2
AC BC
2min
16 x
2
=
2
0
x1 8
684.887 ft
y
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
2
2
4 x 1 8
A 2x y 16
ft
D. 100
2y = 1 + y² y² - 2y + 1 = 0 (y – 1)² = 0 y=1 log x = 1, then x = 10 1 = 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) c om bin( x y) C( 95 5 ) 0.23 P 1 C( 100 5 )
2
x F ind( x) 2 .8 3
513.08 ft
AB 2 453.673 ft
226.836
2x
2
16 x
y 2x
2
A ( x) 2 x 4 x
AB
(x, y)
( maximum)
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 fluid at a rate of 108π m³/sec. How Ho w 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, 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
Pw
1
Ps
4 7 P P s 1 P w 1 P s P w P s P w P
40. This calculates an adjustment 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: A. Transit Rule B. Compass Rule 40
= C. Closure Rule
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 ) 1 6 t 96t 190
D. Closed Traverse Golden Rule
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 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
s 2
W
80 5
8
40 0 kg kg 1 40
9.8 1
g
M g 1 3.7 3 kN kN
m
10° 2 s FR 700N
F
s i n( ) FR 3084.88 N W si
P
F v 6 85 85 52 52. 98 98 W
P
1. 93 93 h p 9 1.
vi 9 0k 0kph t 6s a
200 100 0
0
2
4
6
8
10
t 2
16 t 96t 190 190
5
42. Calculate the power required of a 1400-kg car under the 42. 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 F R = 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
s( t)
0t 6
80
=
300
vf vi t
m
25
vf 110kp 110kph h
s
0.93
m 2
s 1996.3 N
F
M a FR
P P
Fv f 60997.94 W 81.8 hp
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°
0kph 8 0k
v
c
a = 14°
θ
C A
m
30.56 30.56
s
Using Napier’s Rule: Napier’s Rule: sin a = tan Bc tan b sin 14° = tan b/tan 30° b = 7°57’ 7°57’ θ = 121°30’ 121°30’ - 7°57’ 7°57’ θ = 113°33’E
Bc
cc
b
a
b
A c
45. Find the area of the region bounded by the curves
= 12 + 4 the x-axis, x = 1, and x = 4. A. 4 ln 5 B. 4 ln 6 45 f ( x)
C. 6 ln 4 D. 6 ln 5 12x 12 x
x
2
4
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
4 3 2 f( x) 1 0
1
0
1
2
3
5
6
x
4
A
4
12x d x 8 .3 1 8 x2 4 1
6ln( 4 )
8.31 8
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 12 x 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
X L/2 - 1’
1’
L/2
B
A
X + 40
X + 35 L/2 – ½’
L/2
BC let
x = weight of of billy x + 5 = weight wei ght of bobby
L ( x 5) 2 x ( 35
2
=
1
5L 2
x)
L
2
x 5L
CA
5
=
=
=
L x 2
5 =
( 35
x 5) L
2
5 L 40
5L 40 40
L
1 2
16t
2
( 1 2 t ) ( 3 2 1 6t ) 2( 50t 64 ) d v( t ) S( t ) 2 dt ( 16t 32 )
9t
16 v( 1) 5 .6 v( 2) 12
2
t 6 1 2 3
12t
49. Given the sides of a triangle ABC, a = 36.3 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
½’
2
S( t )
36.3 b
sin( B)
=
b
2 3.9
A
77.3 °
a
sin( A ) bsin ( A ) 39.96 a sin 39.96 ° a
B
C
62.74 ° A B 62.74
c
a sin( A )
sin( C)
3 3.0 8
50. The horsepower that can be safely transmitted to a shaft varies jointly as the shaf t’s t’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
=
14 k
k d
3
P
d3 40 3
1
24 1 20 2 1 3 P ( 80 ) 3 90 24
51. A record enthusiast enthusiast decided to calibrate his 33 33⅓ ⅓ 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 4
V ( r)
3
r 3 ( 2r) 3 ( 4r) 3
V ( r) 4
52. How much money must initially be deposited in a savings 52. 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 series factor: (A/G, 5%, 10) = 4.0991 A. Php 31 426.49 B. Php30 504.19 52 P
=
6500 6500 500x 500x
x
1.05
1
x
P 30504 30504.38 .386 6
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
=
3
=
292 r 3
73 r 292 r
V_A ( r)
4 292 r
1 ( 60 ) 2 ( 90 )
2
1 2
x 81. 24 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
4
3
3
3
=
3 73 r 2
3
3
6000 ( 7.7217 7.7217 ) 500 7.7 2 17 4.0991 30504.19
P
R
C. Php28 726.49 D. Php27 029.39
P A i n P i n A i n G A G A
10
3
3
R 3
2
=
4.18
73 r
r 3.000 3.000474 474335 335956 956751 751449 449
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 93000000 mi) 24hr 3600s 365day 1day 1hr
18.529
mi s
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
Let x = speed of the current
[ = ] 72 = 1 1 ( 72 ) 30− 45− 45 − 1.5 1.5 =23030+ ++ = 6 mph
40°
vg
2
2
600 80 2( 80 ) 600 600 cos cos ( 110° 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 0.40x 70 0.40x x 100 x .60 x 3 0 0 .6 60 0 .6 .6 x x 100 x
40 0.40x 0.40x
=
19 26
x 8 .0
d1
1 2
BV 3 5000 50000 00 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 2) 1 80
=
n 2
0.3
2d
L 2 ( 4 4. 7 ) 89 8 9.4 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
=
c
2
=
b b
=
a
6 6
3
e2 c
b
e2 3
2
0.75
2
a
b2 2
28.448
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
6
P 0 .2 5 0.60 0.15
10 3 6 1
P 0.09 0.091 185
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’
0 1 R
2
L = 2d
4
42 5
2
P
3
[ 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
=
D3 d1 d2 d3 377360
16 n 9
L
( 530000 ) 106000
2 5 2 d2 ( 53000 530000 0 10600 106000 0 ) 1 69 6 00 5 2 d3 ( 53000 530000 0 10600 106000 0 169600 ) 101 7 60 5
2
d 4 4.7 m
c
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 3 rd year using MACRS Method. A. P122,640 C. P242,000 B. P128,556 D. P146,000 60
2.4
d 4 6. 2
R
=
( R 10 )
2
L
10’
R
2
2
L/2
2
R
=
2
R
=
L
80
2
5
2
L
80 L
5
( R 1)
=
L
R 2
8
27 L 2
=
L
8
2
27 L 2
L 27 2 2
365
365
80 40
67. The average annual incomes of high school and college 67. 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
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 2
SL
8M R
=
S
8M R
=
L( 2 S L)
=
R
D. 2.391 mi
2100 21000 0 0.8 0.8 35000 35000 0.2 0.2 23800
EXP EXP
R
B. 2.191 mi 71
SL
L( 2 S L) 8M 550 550 ( 2 600 600 550 ) 1117.19 8 40
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
2
x
11 ( 11 x) 23 x 2.091 11
12
=
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 3 2
=
2 88
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. Third Rule. A. 225.2 m2 C. 221.75 m 2 B. 227.15 m 2 D. 222.5 m 2 70 d
h1 2h odd 4h even hn 3 15 A [ 5.59 5.59 2 ( 2.30 2.30 ) 4( 3.38 3.38 3.96 3.96 ) 4.8 ] 3 A
=
A 221.75
71. A road is tangent to a circular lake. Along the road and 12 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 74. 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. 75. 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 )
=
1 0 ( n 1 ) n
n 3 ( 2n 1) 120 10 ( n 1) n 120
