What's the difference?
(Dimensions and Relative Proportions)
(Dimensions and Relative Proportions)
A 6-foot man casts a shadow of 7 feet. At the same time, a tree casts a shadow of 35 feet. How tall is the tree?
Solution:
6/7 = x/35
(35)(6)/7 = x
x = 30
Answer: The tree is 30 feet.
Before practicing, the players jogged around a square park that is 324 square meters. How many times must they jog around that park to cover a distance of 1.08 km?
Solution:
Sqrt(324) = 18
1.08 km x 1000 m/1 km = 1080 m
1080 ÷ (18)(4) = 1080 ÷ 72 = 15
Answer: 15 times
The distance travelled in a given time is directly proportional to the speed. If a car goes 70 km at 30 kph, how far will it go in the same time at 45 kph?
Solution:
D1/S1 = D2/S2
70/30 = D/45
(70)(45)/30 = D
D = 105
Answer:The car could go 105 km at 45 kph.
The color yellow has a frequency of 5 x 10^14 Hz. The light travels through the air at a speed of 2.997 x 10^8 m/s. Determine the wavelength of these light waves.
wavelength = v/f
= (2.997 x 10^8 )/ (5 x 10^14)
= 0.5994 x 10^-6
= 5.994 x 10^-7
Answer: The wavelength of the color yellow is 5.994 x 10^-7 meters
Rey arrived in the Philippines from America. He was informed that the exchange rate for a US Dollar was P50. He had $2250. How much in Pesos was his $2250?
Solution:
1/50 = 2250/x
x = (2250)(5) = 112500
Answer: It’s P112500.
A steel bar weighing 150 kg is divided into 2 parts, so that the ratio of the weight of the lighter part to that of the heavier part is 2:3. How heavy is each part?
(150-x)/x = 2/3
3(150-x) – 2x
450 – 3x = 2x
450 = 5x
x = 450/5 = 90
150 – 90 = 60
Answer: The lighter part weighs 60 kg while the heavier part weighs 90 kg.
The dimensions of a painting are shown as:
Width: 3x + 2
Height: 2x
Find the area in terms of x
Solution:
Area = length x width
A = (3x + 2)(2x)
A = 6x^2 + 4x
If an athlete drinks 6 glasses of water every game, at least how many liters of water should he bring per game if one glass is 220mL?
Solution:
1 L = 1000 ml
220 x 6 = 1320 ml
1320 ml x (1 L/1000 mL) = 1.32
Answer: at least 1.32 liters of water
Dolphins use echolocation to navigate and hunt. Determine the time delay between the sending of a pulse and the return of its reflection from an object located 375 m away. Approximate the speed of the sound waves as 1500 m/s.
Solution:
Sound travels forward and the back so
d = (2)(375) = 750
t = d/s
t = 750/1500
t= 0.5 s
Answer: The total time it takes the pulse to hit the object and return is 0.5 seconds
One minus three times the reciprocal of a number is 4.
Find the number.
Solution:
1 – 3/x = 4
x( 1 – 3/x = 4)
x-3 = 4x
-4x+x = 3
-3x = 3
x = -1 Answer: The number is -1.
A boy and a 41.9 kg girl, both wearing roller skates face each other at rest on a roller rink. The boy pushes the girl, sending her eastward with a speed of 5.62 m/s. Meanwhile, the boy ends up rolling backward with a speed of 3.87 m/s Neglecting friction, determine the mass of the boy.
Solution:
Total momentum is 0 so the magnitude of p(boy) and p(girl) are equal.
p=mv
Let x = mass of boy
x(3.87) = (41.9)(5.62)
x = (41.9)(5.62)/(3.87)
x = 60.84702842 or 60.85
Answer: The mass of the boy is 60.85 kg
Find x that will make the angles complementary if one angle is 3x^2 + 11 and the other is x^2 +15.
Solution:
(3x^2 + 11) + (x^2 + 15) = 90
4x^2 + 26 = 90
4x^2 = 90-26
4x^2 = 64
x^2 = 16
x = 4
Answer: For the angles to be complementary, x = 4
The speed of light is 3 x 10^8 meters/sec. If the sun is 1.5 x 10^11 meters from the earth, how many seconds does it take for sunlight to reach the earth? Express the answer as a scientific notation.
Solution:
Speed = Distance / Time
3 x 10^8 = 1.5 x 10^11 x Time
Time = (1.5 x 10^11) / (3 x 10^8)
= 0.5 x 10^(11-8) = 0.5 x 10^3 sec
Answer: It takes 5 x 10^2 seconds for sunlight to reach the earth
The formula h = 19.6t + 1/2 (-9.8)t^2 approximates the height, in meters, of a ball thrown upward after t seconds. How high will the ball be after 3 seconds?
Solution:
h = (19.6)(3) + 1/2(-9.8)(3^2)
h = 58.8 + (-4.9)(9)
h = 58.8 - 44.1
Answer: The ball will be 14.7 meters up in the air
The product of 2 rational numbers is 9.6. If one of the rational numbers is 66/7, find the other rational number, as a fraction in the simplest form.
Solution:
(66/7)(n) = 9.6
n = (96/10)/(66/7)
n = (96/10) x (7/66) = (8/5)(7/11) = 56/55
Answer: The other rational number is 56/55 or 1 & 1/55
Television screens are usually measured by the length of its diagonal. The television set has a 60-inch diagonal. The screen is 12 inches wider than its height. Find the dimensions of the screen.
Solution:
height = x inches
width = x + 12 inches
diagonal = 60 inches
(use the Pythagorean Theorem)
a^2 + b^2 = c^2
x^2 + (x+12)^2 = 60^2
x^2 + x^2 + 24x + 144 = 3600
2x^2 + 24x = 3456
x^2 + 12x = 1728
x^2 + 12x + 36 = 1728 + 36
(x+6)^2 = 1764
x + 6 = +/- 42 | x = 42-6 or x = -42-6
x = 36 or -48
x = 36 (A physical object only has positive dimensions)
Answer: The height of the television is 36 inches. The width of the picture is 36 + 12 or 48 inches.
Find the value of x that will make □MATH a square if diagonal MT = 6x^2 - 1 and diagonal AH = 3 -3x^2
Solution
MT = AH
6x^2-1 = 3-3x^2
6x^2+3x^2 = 3+1
9x^2 = 4
x^2 =4/9
x = +/- 2/3
Answer: x = 2/3
A student jogs from home to a convenience store nearby. He runs at 2 m/s for 20 seconds and then at 2.5 m/s for 40 seconds. What was the average speed of that student going to the store?
d = vt
d1 = (2)(20) = 40
d2 = (2.5)(40) = 100
d(total) = 100 + 40 = 140
v = 140/60
v = 2.33m/s
Answer: The average speed of the student is 2.33 m/s
The mass of an astronaut on the moon is 78 kg. Determine the weight of that man on the moon where the gravitational acceleration is one-sixth that of the Earth.
Solution:
gravity on the moon = 9.8/6 = 1.63 m/s
weight = ma
= 78 x 1.63 = 127.4
Answer: The weight of the astronaut is 127.4 N
A number increased by 16 times its reciprocal is equal to 10. Find the number/s
Solution:
x+16(1/x) =10
x[x+16(1/x) =10]x
x^2 + 16 - 10x = 0
x2 -10x + 16 = 0
(x-8)(x-2) = 0
x=8 or x=2
Answer: The numbers are 8 or 2.
There are 2 train cars of different masses on a track. One car with a mass of 1500 kg is at rest while the other is moving towards the other car at a speed of 72 kph. The moving car collides into the other car and the both end up moving forward together at a speed of 12 m/s
Solution:
Let x be the mass of the moving car
p(before) = p (after)
p =mv
72 kph x (1000m/km) x (1 hr/3600 s) = 20 m/s
x (20) = x (12) + (1500)(12)
20x = 12 x + 18000
8x = 18000
x = 2250
Answer: The moving train car has a mass of 2250 kg
There is an equilateral triangle with all sides having a measurement of 36 cm. We have line x bisecting the triangle right down the middle. Find the length of line x.
Solution:
All angles of an equilateral are equal and their sum = 180. Therefore each angle of the triangle is 60 degrees.
sin 60 = x/36
x = 36 sin 60
x = 36 sqrt(3)/2
x = 18 sqrt(3) or 31.18
Answer: Line x is 31.18 cm
A marketing executive travelled 810 km on a corporate jet in the same amount of time that it took to travel an additional 162 km by helicopter. The rate of the jet was 360 kph greater than the rate of the helicopter. Find the rate of the jet. Hint: Rate = Distance/Time
Solution:
Since Rate = Distance/Time, Time = Distance/Rate
810/(r+360) = 162/r
(r)(810)/(r+360) = 162
810r = 162(r+360)
810r-162r = 58320
648r = 58320
r = 58320/648 = 90
90+360 = 450
Answer: The rate of the jet is 450 kph.
The diameter of the sun is 8.656 x 10^5 miles. Using 1 mile = 1.61 km, what is the diameter of the Earth in kilometres if the sun’s diameter is 109 times that of the earth? Round off answer to the nearest whole number.
Solution:
(8.656 x 10^5 x 1.61)/109 = 0.1278546 x 10^5
= 1.278546x 10^4 = 12785.46 = 12785
Answer: The diameter of the Earth is 12785 km
Two college students have started their own business building computers from kits. Working alone, one student can build a computer in 20 hours. When the 2nd student helps, they can build a computer together in 7.5 hours. How long would it take the 2nd student, working alone, to build the computer?
Solution:
7.5/20 + 7.5/c = 1
20c (7.5/20 + 7.5/c = 1)
7.5c + 150 = 20c
150 = 12.5c
150/12.5 = c
c = 12
Answer: It would take 12 hours for the 2nd student, working alone, to build the computer.