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2012 2013 UIL Calculator Study List Problems
09A-36. The half life of Uranium 230 is 20.8 days, the time needed
for 50% to decay. How long would it take for 20% of Uranium 230
to decay? --------------------------------------------------------------------- 36= ________________ days
At = A0ekt
k = 12ln
20.8 .8 = 1 e
kt t =
12
20.8ln(.8)
ln = 6.70
09A-38. If x2 + y
2 = 152 and x
2 – y
2 = -49, what is the smallest
value of (x + y)2 ? ---------------------------------------------------------- 38= ____________________
Add equations → 2x2 = 103 x2 = 103/2 x = 103/ 2
y2 = 100.5 y = 100.5
choose x = 103/ 2 y = 100.5 (x + y)2 = 8.11
*** TI 89 solver Look at the different pairs and pick the correct pair.
09F-37. A pipe has an outside diameter of 1.25 in and an inside
diameter of 0.75 in. If Kelly hacksaws the pipe in two, what
fraction of the pipe cross sectional area is sawn when the blade
breaks through to the inside? --------------------------------------------- 37= __________________ %
cos α = .75 / 1.25 α = 53.13
θ = 2α = 106.26 → 1.85… (rad)
r1 = .75/2 , r2 = 1.25/2
22
2
2 22 1
( sin )100%
r
r r
= 22.2
09G-36. In an egg toss, a strategy is to throw the egg with the
lowest possible velocity but still cover the distance to the catcher.
If the catcher is 25 ft from the thrower, what is this velocity? ------- 36= _________________ ft/s
Lowest V occurs when θ = 45°
max
2 sin 2h
vd
g
2(1)25
32.17
v v = 28.4
25
16
1
8
25
16
9
9
Page 2
09I-36. A dog is tied to a 16 ft square shed with a 25 ft long leash.
What is percent difference in the dog’s roaming area if it is tied in
the middle of one side or on a corner? ---------------------------------- 36= __________________ %
½ (25)2 + ½ (17)
2 + ½ (1)
2 = 1437.278…
¾ (25)2 + ½ (9)
2 = 1599.856…
1437.278…, 1599.856…, % chg = 11.3
10D-37. A major league baseball pitcher throws a fast ball at
98.5 mph. If the release is 5 ft 6.7 in above the ground and
horizontal, how far would the ball travel unencumbered before
hitting the ground? --------------------------------------------------------- 37= ______________ ft(SD)
Vertical: y = y0 + v0t + ½ at2 but y and v0 = 0 so -2y0 / a = t
2
t =
06.72 52y 12
a 32.17
= .5878…
Horizontal: v = 98.5 (88/60) = 144.466.. ft/s
x = x0 + v0t + ½ at2 but x0 and a = 0 so x = (144.466…)(.5878…)
x = 84.9 {3SD}
10D-38. A red car sitting still is passed by a green car traveling at
a velocity v. After a 1.35 sec delay, the red car accelerates at
10.13 mph/sec and catches up with the green car in 13.57 sec.
How fast was the green car going?--------------------------------------- 38= ____________ mph (SD)
d = d
½ at12 = vt2
½ (10.13)(13.57)2 = v(13.57 + 1.35)
4SD 4SD 4SD
v = 62.51 {4 SD}
Page 3
10G-36. What is 349,441-902,521
? ---------------------------------------- 36= ____________________
-902,521 log 349,441 = -5003011.31907
Use the integer part less 1 for the new exponent: 10-5003012
Add +5003012 to what’s in the calculator: .68093 → 10.68093
= 4.80
= 4.80 10-5003012
10H-36. Amy can bike to the store in 7 min, and walking takes 25
min. What percent of the distance to the store was traversed by
bike if she had a flat along the way, and the total travel time was
12 minutes? ----------------------------------------------------------------- 36= __________________ %
12 = x (1 x)
1 17 25
x = .722 .722 100% = 72.2
or (1/7)t + (1/25)(12 – t) = 1 t = 91/18 (1/7)(91/18) = .722 100% = 72.2
10H-38. In Seguin, Texas one day the high was 101°F at 5 PM and
the low was 76°F at 5 AM. Assuming the temperature varied
sinusoidally, how many hours after 9 AM did the temperature
first hit 93°F? ---------------------------------------------------------------- 38= __________________ hr
T(t) = A cos (2π(f)(t)) + D
A = amplitude (H – L)/2 (101 – 76)/2 = 12.5 → -12.5
{must be negative, since we started with the high}
f = frequency (1/period) (1/24)
D = vert. displacement or centerline (76 + 101) / 2 = 88.5
93° = -12.5 cos (πt / 12) + 88.5 t = 7.4066… hours
5 AM + 7.4066797 – 9 AM = 3.41
10I-36. Two integers, each greater than 100, multiply to 46,620.
Both are divisible by 6, and the smaller number is divisible by 5
as well. What is the sum of the two integers? ------------------------ 36= ______________ integer
46620 = 215.916…
215 is divisible by 5.
However, it does not divide evenly into 46620 and is not divisible by 6..
Try next lower number divisible by 5 → 210. This is also divisible by 6.
This does divide evenly into 46620. → 222. This is also divisible by 6.
210 + 222 = 432
8d
12
18
12
3960
3960
x
Page 4
10I-37. What is the minimum elevation above the earth’s surface
necessary to have a direct line of sight to both Fiji and New Zealand
if they are 1314 mi apart? Neglect refraction. -------------------------- 37= _________________ mi
arc length = r θ 1314 = 3960(θ) θ = 19.011…°
cos (19.011…° / 2) = 3960 / (3960 + x)
x = 55.1
11B-38. What is the electrical resistance of an immersion hot water
heater necessary to bring 8-oz of water from room temperature, 25°C, to
its boiling point, 100°C, in 60 sec? The specific heat of water is 4.19
J/gK. Power generated by the heater in watts (= J/s = Vamp = V2/ohm)
is the product of the voltage, 110 V, and the current in amps. The
current is the voltage divided by the resistance in ohms. ------------------ 38= _____________ ohm
Q = energy P = power V = voltage I = current
R = resistance T = temperature t = time m = mass c = specific heat
Q = mcT P = VI P = Q / t Q / t = VI Q = VIt I = V/R
mcT = VIt = V(V/R)t
R = (V2
t) / (mcT) and m = 8 oz.(liq) = 236.588 ml = 236.588 g
R = (1102(60)) / (236.588)(4.19)(100 – 25) = 9.76
11D-38. Mike and Mary stand unmoving 24 ft apart. Ned stands 18 ft from
Mike and 8 ft from Mary, forming a scalene triangle. What is the
shortest distance Ned can move to create a right triangle? ---------------- 38= _______________ ft
If let “d’ be the median to the 24 foot side and use the formula
for finding median lengths in triangles:
2 2 22
2c
a b cm
; where a + b > c.
d = ½ 2 2 22(18 8 ) 24 d = 7.071… 12 – d = 4.93
167
S95
A
C
D
Page 5
11E-26. Harvey in Aspermont, TX wants to arrange a meeting with Mike
in Childress which is 95 mi north, and also with Harry who lives in
Decatur, 17 mi due east. If the meeting site selected was Seymour
which is equidistant from all three towns, how far does each person
travel? ---------------------------------------------------------------------------- 26= ______________ mi
S is located at the circumcenter of a right triangle.
2 295 167
2
= 96.1
11E-38. The distance x a carbon atom moves in solid iron equals
Q0 RT2D t exp where D0 = 0.2 cm2/s, t = elapsed time, Q = 32,000 cal/mole, R = 1.987 cal/(mole-K) and T = absolute temperature (K).
Carbon is placed on the surface of iron powder 55m in diameter.
Calculate the minimum temperature necessary to diffuse the carbon to the
center of the iron particles in 3 hr.? ------------------------------------------- 38= ______________ °C
(55 10-4
cm) / 2 = .00275
.00275 = 32,000
1.9872(.2)(3 3600)
Te
Solve: T = 798.7117…K 798.7117… – 273.15 = 526
11F-37. A kid used a clothes pin to attach a playing card to a spoke
on the front wheel of his bike. Once every wheel rotation, the card
“clicked” when it struck the bike frame yoke. If the front wheel has an
18-in diameter, what is the bike velocity when clicks are ¼ s apart? ---- 37= _____________ mph
(18π in/rev) (4 rev/s) (1 ft/12 in) (1 mi/5280 ft) (3600 s/hr) = 12.9
11G-26. A surveyor estimated a rectangular field to be 2.93 acres. The
subsequent survey produced side dimensions of 221.52 ft and 589.91 ft.
What is the percent error of the estimate? ----------------------------------- 26= ___________ %(SD)
A = (221.52)(589.91) = 130,676.86…
A (640 acres / 52802) = 2.99992… = B
[(2.93 / B) {3SD} – 1] {2SD} 100% = -2.3
or
B, 2.93, % chg = -2.3
y = b
r
ds
Page 6
11G-37. The lines y = 12x, y = 4x+8 and y = b intersect to form a
triangle. Solve for b if the length of the line segment associated with
y = b equals 15 and b is negative. --------------------------------------------- 37= _________________
x2 – x1 = 15
y / 12 – (y – 8) / 4 = 15
y = -78.0
or
12x = 4(x – 15) + 8
x = -6.5
y = 12(-6.5) = -78.0
11H-37. It takes Abe 30 min to row his boat 2 mi up river. He rows back
to where he started in 10 min. What is the river velocity? ---------------- 37= _____________ mph
(b – c) (30/60) = 2 (b + c)(10/60) = 2
Solve: b = 8 c = 4.00
11I-38. Marin runs a mile in 5 min 30 sec. She and Mary, who runs a
mile in 7 min 26 sec, start at the south end of a ¼ mi circular track.
Marin takes off running on the track, and at the same time, Mary takes
off in a straight line off the track, meeting up with Marin before she
finishes one lap. Considering due east to be 0°, at what positive angle
does Mary need to run? --------------------------------------------------------- 38= ______________ rad
rt = d
t/446 = d t/330 = s t = 330s
sin (θ/2) = d/2r s = θ r θ = s/r
d/s = 330 / 446 d = 330 s / 446
nsolve sin (x/2) = 330x / [446(2)] x = 2.60706487 rad
(π – x)/2 = .267263892 π /2 - .2672 = 1.30
12B-37. Bulk potatoes are sliced into 0.02 in thick slices to make potato chips. A potato loses 75% of its mass during frying. Potatoes on average are 6 in long, weigh 0.8 lb and cost $0.88/lb. An 11-oz bag of chips costs $2.58 and has 128 chips. What is the ratio of the cost of chips to the cost of the starting bulk potatoes needed to make those chips (on a mass basis)? ------------------------- 37= ___________________ 6 / .02 = 300 slices 128/300 × (.8 lb) × ($.88/lb) = $0.30037… 2.58 / .30037… = 8.59
12
3
6
9
Page 7
12D-36. How long after 5:30 do the minute and hour hands of a clock first align? ------------------------------------------------------------------------------------- 36= ___________________ min 11/12 T = 30 + 27.5 T = 62.7 Alternate Solution:
3011 30 3060
5.5 / min
2D-38.One thousand synchronized lights are place in a line spaced 1 ft apart. The first light flashes on/off, then after a time delay τ, the second light flashes similarly, then the third, etc., such that the light blip “moves” along the row with constant velocity. What is τ if the apparent blip velocity is 1.5 times the speed of light, 186,000 mi/s? ------------------------------------------------------------------- 38= __________________ μsec 1.5(186,000) t = 1 / 5280 (10
6) t = .000679
12E-37. It takes 1.5 J (= 1 Nm) of energy to knock over a bowling pin and move it out of the way. What is the minimum velocity of a 15-lb bowling ball to bowl a strike, knocking over all 10 pins? The bowling ball energy is 0.5 mv
2 where m is
the ball mass and v is its velocity. 1 N = 1 kgm/s2. ------------------------------ 37= ___________________ fps
15 lb → 6.80388… kg (HP conversion) 10(1.5) = ½ (6.80388..)v
2
v = 2.09982… m/s × 100 cm/m × 1 in/ 2.54 cm × 1 ft/12 in. = 6.89
12F-37. In Olympic archery, the archer shoots an arrow with an average speed of 320 ft/s at a target 70 meters away. Assuming the arrow is released at an elevation equivalent to the target’s bulls eye, what should the archer’s release angle be (positive, less than 45° with 0° parallel to the ground)? ------------ 37= ___________________ deg 70 m × 100 cm/ 1 m × 1 in/ 2.54 cm × 1 ft/ 12 in. = 229.658… ft {A} dh max = (v
2 sin 2θ) / g {A} = 320
2 sin 2θ / 32.17 θ = 2.07
12G-36. Sam begs for bread from a local baker. The first day, Sam received a full loaf of bread. On the second and subsequent days, Sam got 40% of the preceding day’s allocation. How much total bread will Sam ultimately receive? 36= _____________ loaves S = 1 / (1 -.4) S = 1.67
12H-37. A chair was caught up in a tornado, rotating around a 1500 ft circumference with a speed of 110 mph. After picking up the chair, the tornado traveled 5 mi at 35 mph before dissipating. How far did the chair travel? - 37= ____________________ mi 5/35 = 1/7 hr 110(1/7) + 5 = 20.7
Page 8
12I-36. Sam makes coffee by adding one teaspoon of coffee concentrate to 6 oz of water. How much coffee concentrate is needed to make 25 gallons of coffee? 36= ______________ qt
(1 / 6)
6 (1 / 6) 25(128)
x
x = 86.486… oz x / 32 oz = 2.70
12I-38. Sonya drove 30% of the distance to her destination at 56 mph. She then sped up so her total average trip velocity was 63 mph. What was her velocity on the second leg of the trip? ----------------------------------------------- 38= __________________ mph
Let distance = 63 mi. .3(63) .7(63) 63
56 63V V = 66.6
sin 82 / 389 = sin A / 182
A = 27.6
180 - (82 + 27.6) = B = 70.4
sin 70.4 / 518 = sin x / 389
x = 45.0
tanA = 1
2 3 2 3
x
x 16.102
tanB = 30
tanC = 180 - (A + B) = 133.89…
½ bh - 2 sin sin
2sin
c A B
C = shaded area
½ (x)( 2 3x ) - 2( 3) (sin16.102)(sin30
2(sin133.89...)
x = .904
2
2 23 (sin16.102)(sin30)3 (1.44...) .9042(sin133.89...)
xx x
x = .7914… 2x = 1.58
04A-69.
04A-69=_______________________________
SCALENE TRIANGLES
82
182
389 518
deg?
REGULAR HEXAGON
04B-70=_______________________________
04B-70.
Shaded Area = 0.904
?
midpoint
center
Page 9
Use the small similar triangle to find the angle:
Tan θ = (5.69 – 4.67) / 1.44 = 1.02 / 1.44
θ = .616
w2 = (2.14)
2 + (4.05)
2 – 2(2.14)(4.05) cos .87
w = 3.131242
sin .87 / w = sin θ / 4.05 = 1.4196 rad
But 1.4196 rads is too small ( < /2).
This is an example of the ambiguous case for a SSA
triangle.
So: 1.4196 = 1.7219658 rads =
θ = 1.7219658 rad
β = π – (.65 + θ)
β = .7696268 rad
α = π – (1.41 + β)
α = .9619658 rad
sin α / z = sin 1.41 / w
z = 2.60218
x2 = 3.02
2 + z
2 – 2(3.02) z cos (.65)
x = 1.84
05C-60.
05C-60 = _______________________
CIRCULAR ARC, RIGHT TRIANGLES
rad? 1.44
5.694.67
rad? 1.44
5.694.67
5.69 - 4.67
06E-60.
06E-60. = _______________________
SCALENE TRIANGLES
?
4.051.41 rad
0.65 rad 0.87 rad
3.02 2.14
x
4.051.41
.65 .87
3.02 2.14
zw
A
1- A
1
Page 10
1.72 = 2.18
2 + 1.18
2 – 2(2.18)(1.18) cos A
A = 50.7549…
B = A + 21.5 = 72.2549
x2 = 2.18
2 + 1.82
2 – 2(2.18)(1.82) cos B
x = 2.38
Let side = 1
A = ½ bh
A = ½ (1)(1 - A ) Use TI solver.
A = .25 → A = .5
tan θ = (1 - .5)/ 1 = ½
θ = 26.6
07E-60.
SCALENE TRIANGLES
07E-60 = _______________________
2.18
?
1.70
1.82
1.18
21.5o
08C-60.
08C-60 = _______________________
AREA(TRIANGLE) = A
SQUARE AND RIGHT TRIANGLE
deg?
A
?
747 747
r
60
747 747
w
508
830
992
h
508
992
xw
x
Page 11
360 (60 )
360
π(747)
2 = 1.33 10
6
θ = 26.87...
cos θ = w / 747
w = 666.326…
2w = 1332.65…
2w – 747 = 586
5082 = 508
2 + 992
2 – 2(508)(992) cos x
x = 12.478…
tan x = w/ 830 w = 183.67…
h = 2w = 367.35…
tan θ = 367.35… / 992 θ = .355
08E-60.
08E-60 = _______________________
?
AREA(SECTOR) = 1.33X106
SECTOR, EQUILATERAL AND ISOSCELES TRIANGLES
747
09F-60.
09F-60 = _______________________
RECTANGLE AND ISOSCELES TRIANGLES
AB is a straight line segment
508 B
A
830 992
rad?
x
14o
r
w
2
Page 12
πr2/2 = ½ bh b = r
πr2
= rh
h = πr
let r = 1 h = π
tan θ = π/2
θ = 57.5
w = r tan 14°
height = r
rect = wr = r2 tan 14°
570 = ¾ πr2 + r
2 tan 14°
r = 14.8
57,200 = πR
2 – π(R – 91.5)
2
R = 145.2…
r = 145.2… - 91.5 = 53.7
10D-60.
THREE-QUARTER CIRCLE AND RECTANGLE
Total Area = 570
10D-60 = _______________________
R = ?
14.0o
10F-60.
10F-60 = _______________________
Hatched Area = 57,200
CONCENTRIC CIRCLES
R = ?
91.5
10A-60.
RIGHT TRIANGLE AND SEMICIRCLE
10A-60 = _______________________
Semicircle Area = Right Triangle Area
Radius
deg?
3.89
2.05
2.05
d
2.05d
Page 13
Vc - Vf = Volume
V = π(r)2h – 1/3 π (r1
2 + r2
2 + r1r2)h
V = π(90.5/2)2(88.8) – 1/3 π [(90.5/2)
2 + (40.5/2)
2 +
(90.5/2)(40.5/2)]88.8
V = 257,000
d2 + 2.05
2 = 3.89
2
d = 3.3059…
tan θ = 2.05/(2.05 + d) θ = 20.9°
11B-49.
CYLINDER WITH FRUSTUM CAVITY
11B-49 = _______________________
Volume = ?
88.8
40.5
90.5
11D-60.
SQUARE AND RIGHT TRIANGLES
11D-60 = _______________________
3.89
2.05
deg?
L 86.4
44.8
w
419 - x
Page 14
176 – 89.6 = 86.4
44.82 + 86.4
2 = L
2
L = 97.32420…
A = ½ (89.6)L = 4360.12…
TA = 4A + 5(89.6)2
TA = 57,600
V = (1/3)BH B = ½ bh
V = (1/3)(1/2)(419 – x)(w)(h)
V = [(419 – x)(w)(h)] / 6
(419 x)(w)(h)
6(419 x)(w)(h)
419(w)(h)6
= .1
419 x
(419 x)6 419
6
= .1 Solve x = 190
11E-49.CUBE AND PYRAMID
Total Surface Area = ?
11E-49 = _______________________
89.6
176
11G-50.
TRUNCATED RECTANGULAR SOLID
?
11G-49 = _______________________
419
Missing Corner VolumeTruncated Solid Volume
= 0.1
Page 15
40.92 = 44.3
2 + 18.1
2 – 2(44.3)(18.1)cos θ
θ = 67.360…
sin θ / 40.9 = sin / 18.1
= 24.107…
½ (18.1)(x)sin θ = ½ (40.9)(44.3 – x)sin
x = 22.15
AB2 = x
2 + (18.1)
2 – 2(x)(18.1)cos θ
AB = 22.6
Let radius of smaller sphere = 1. (4/3)πr
3 = 9.56 (2/3)π(1)
3
2r
3 = 9.56 r = 1.6845…
Sin θ = r/(r+1) θ = 38.9
11I-60.
11I-60 = _______________________
SCALENE TRIANGLES WITH EQUAL AREA
AB = ?
40.9
A
18.1
B
44.3
12A-49.
SPHERE AND HEMISPHERE
deg?
12A-49 = _______________________
Volume (Sphere) =9.56 Volume (Hemisphere)
r
r + 1
θ
12A-60.RECTANGLE
AB = BC = 0.539 BD = ?
12A-60 = _______________________
0.194
0.468
A
B
CD
Page 16
AC
2 = .194
2 + .468
2 AC = .5066….
BC2 = AC
2 + AB
2 – 2(AC)(AB) cos BAC
BAC = 61.9683…
tan CAD = CD / AD
CAD = 67.4845…
BAD = BAC +CAD = 129.4528…
BD2 = AD
2 + AB
2 – 2(AD)(AB) cosBAD
BD = .679
V = πr
2h + (1/3)π r
2 h
V = (4/3)π r
2 h
.0634 = (4/3)π (.331/2)
2 h
h = .553
12E-49.SLANT CIRCULAR CYLINDER AND CONE
Combined Total Volume = 0.0634
12E-49 = _______________________
?
0.331 0.331
560b
45o
60o
75o
560 2560 2
Page 17
cos 75° = b / 560 2 b = 204.974… A = ½ ab sin C
A = ½ (560 2 )(204.974…) sin 75°
A = 78,400
12I-60.SQUARE, EQUILATERAL AND RIGHT TRIANGLE
12I-60 = _______________________
Hatched Area = ?
560