Download English Measurement Relationships

FLC
Ch 10
Math 30 Prealgebra
Sec 10.1: Using Unit Fractions with U.S. and Metric Units
Defn
A unit fraction is a fraction that shows the relationship between units and is equal to 1. Ex
English Measurement Relationships
Length
Weight
1 foot (ft) = 12 inches (in)
1 yard (yd) = 3 feet (ft)
1 mile (mi) = 5280 feet (ft)
1 pound (lb) = 16 ounces (oz)
1 ton (T) = 2000 pounds (lb)
Capacity
Time
1 cup (c) = 8 fluid ounces (fl oz)
1 pint (pt) = 2 cups (c)
1 quart (qt) = 2 pints (pt)
1 gallon (gal) = 4 quarts (qt)
1 minute (min) = 60 seconds (sec)
1 hour (hr) = 60 minutes (min)
1 day = 24 hours (hr)
1 week (wk) = 7 days
Converting Among Measurement Units
1) Multiply when converting from a __________ unit to a ___________ unit. ex
2) Divide when converting from a __________ unit to a ___________ unit.
Weight
1 kilogram 𝑘𝑔 = 1000 grams
1 gram 𝑔 = the basic unit
1 milligram 𝑚𝑔 = 0.001 gram
Length
1 kilometer 𝑘𝑚 = 1000 meters
1 meter 𝑚 = the basic unit
1 centimeter 𝑐𝑚 = 0.01 meter
1 millimeter 𝑚𝑚 = 0.001 meter
kilo
hecto
deka
Volume
1 kiloliter 𝑘𝐿 = 1000 liters
1 liter 𝐿 = the basic unit
1 milliliter 𝑚𝐿 = 0.001 liter
gram
Liter
meter
basic unit
These prefixes identify
units that are larger than
the basic unit.
deci
Meaning
Symbol
kilometer
1000
meters
km
hectometer
100
meters
hm
dekameter
10
meters
dam/Dm
milli
These prefixed identify
units that are smaller than
the basic unit.
Move LEFT
Prefix
centi
kilo
hecto
deka
Basic Unit
deci
centi
milli
Move RIGHT
meter
1
meter
m
decimeter
of a
centimeter
of a
millimeter
of a
meter
dm
meter
cm
meter
mm
Page 1 of 14
FLC
Ex 1
Ch 10
Convert.
a)
b)
84 in =___________ ft
87 in =___________ ft
c)
d)
e)
f)
11 days =___________ hours
192 oz =___________ lb
7 gal=___________ qt
2.25 lb =___________ oz
Ex 2
Judy is making a wild mushroom sauce for pasta tonight for a group of friends. She bought 26
ounces of wild mushroom at $6.00 per pound. How much did the mushrooms cost?
Ex 3
Convert.
a)
b)
c)
44 cm =___________ mm
7.6 km =___________ m
5261 mL =___________ kL
d)
e)
f)
5.9 kg =___________ mg
18 mL =___________ L =___________ kL
12 DL =___________ L
Page 2 of 14
FLC
Ch 10
g)
h)
i)
0.84 cm =___________ m =___________ km
6 mi =___________ ft
6 tons =___________ lb
j)
k)
l)
210 lb =___________ T
15 ft =___________mi
3 oz =___________ T
Ex 4
A dam is 335 meters high.
a) How many kilometers high is the dam?
b) How many centimeters high is the dam?
Sec 10.2: Converting Between the U.S. and Metric Systems
U.S. to Metric
Units of length
1 mile
Metric to U.S.
1.61 kilometers
1 kilometer
1 yard
0.914 meter
1 meter
3.28 feet
1 foot
0.305 meter
1 meter
1.09 yards
1 inch = 2.54 centimeters* 1 centimeter
Units of volume
Units of weight
0.62 mile
0.394 inch
1 gallon
3.79 liters
1 liter
0.264 gallon
1 quart
0.946 liter
1 liter
1.06 quarts
0.454 kilogram
1 kilogram
1 pound
1 ounce
28.35 grams
1 gram
2.2 pounds
0.0353 ounce
*Exact value
Page 3 of 14
FLC
Ex 5
Ch 10
Perform each conversion.
Round your answer to the nearest hundredth if necessary.
a) 12 km to mi
b) 35 m to ft
c) 19.6 cm to in
d) 6.5 L to qt
e) 6 kg to oz
f) 142 cm to ft
g) 60 kph to mph
h) 40 mph to kmp
i)
j)
k)
l)
to Celsius
to Celsius
to Fahrenheit
to Fahrenheit
Page 4 of 14
FLC
Ch 10
Ex 6
Solve. Round your answer to the nearest hundredth when necessary.
a) The average weight for a 7-year-old girl is 22 kilograms. What is the average weight in pounds?
b) A surgeon is irrigating an abdominal cavity after a cancerous growth is removed. There is a supply of
3 gallons of distilled water in the operating room. The surgeon uses a total of 7 liters of the water
during the procedure. How many liters of water are left over after the operation?
c) (calculator) While panning in a river, a prospector found a gold nugget that weighed 2.552 oz. How
many grams did the nugget weigh?
Page 5 of 14
FLC
Ch 10
Sec 10.3: Angles
The word for geometry comes from the Greek words for measure and earth. This is because geometry was
originally used to measure land. Today we use geometry in many fields such as physics, drafting, art,
engineering, and medicine.
Terminology and Symbols
 point

line

line segment

ray

angle

sides of angle

vertex of angle
We use degrees to measure the amount of opening of an angle. (Another form of measurement is
radians.)
Ex 7
Draw each and label the degrees.
a) an angle that is a complete revolution
c) an angle that is one-fourth a revolution
Ex 8
b) an angle that is one-half a revolution
The sides of the angle form ______________ lines. What symbol is used?
Give four different names for the angle.
A
R
𝑥
𝑥
B
T
S
An angle that measures
D
𝑦
C
is called a straight angle. An angle whose measure is between
an _______________ angle. An angle whose measure is between
and
and
is called
is called an _______________
angle. Consider two intersecting lines. Two angles opposite of one another are called _______________
angles. Two angles that share a common side are called adjacent angles.
Page 6 of 14
FLC
Ch 10
Draw two parallel lines. What symbol is used?
A line that intersects two or more lines at different points is called a transversal. Draw a transversal.
Alternate interior angles are two angles that are on opposite sides of the transversal and between (inside) the
other two lines. Identify the alternate interior angles.
Corresponding angles are two angles that are on the same side of the transversal and are both above (or below)
the other two lines. Identify the corresponding angles.
Parallel Lines Cut By a Transversal
If two parallel lines are cut by a transversal, then the measures of corresponding angles are equal and the measures of
alternate interior angles are equal.
Ex 9
measures
.
a) Find the supplement of
Ex 10
.
Find .
b) Find the complement of
Ex 11
Find the measure
and
.
.
𝑦+
𝑦
Ex 12
3𝑥 +
Find .
2
3
3
𝑥−
2
Page 7 of 14
FLC
Ch 10
Sec 10.4: Square Roots and the Pythagorean Theorem
List perfect squares.
Know these perfect squares for test.
Defn The square root of a number
if
= .
√
is the number
=
where
and
. In symbols, √ = ,
- indicates square root and is called a radical sign
√
Note: The result upon taking the square root is always nonnegative.
=√ .
Ex 13
Find , where
Ex 14
Simplify. (Find the square root and simplify.)
a)
b)
√2
c)
√
√ 2
e)
f)
√
− √
3
Ex 15
d)
√
−√
2
−( ) √
3
Find the length of the side of a square that has an area of 16 square feet.
Page 8 of 14
FLC
Ch 10
Ex 16 You can use your calculator to estimate √2 . Without your calculator, decide which two
consecutive whole numbers √2 is between.
Pythagorean Theorem
In any right triangle, if 𝑐 is the length of the hypotenuse and 𝑎 and 𝑏 are the lengths of the two
legs, then 𝑎 + 𝑏 = 𝑐 .
𝑐
𝑏
𝑎
Ex 17
Find the unknown side of each right triangle using the given information.
round answers to the nearest thousandth if necessary.
a)
b)
=
and
Ex 18
= 2
Give the exact answer then
c)
=
=
=
=
Find the area of the shape below made up of a square and right triangle.
10 in.
6 in.
Page 9 of 14
FLC
Ch 10
Ex 19 Barbara is flying her dragon kite on 32 yd of string. The kite is directly above the edge of a pond.
The edge of the pond is 30 yd from where the kite is tied to the ground. How far is the kite above the
ground?
ground
Sec 10.5: The Circle
Defn A circle is a figure for which all points on the figure (circle) are at an equal distance form a given point.
This given pint is called the center of the circle. The radius is a line segment from the center to a point on the
circle. The diameter is a line segment across the circle that passes through the center.
Defn
The distance across the rim of a circle is called the circumference, C. Given any circle, if we take it’s
circumference and divide it by its diameter, we get  . (Greek letter) That is,
C
  and  is a number
d
that’s approximately equal to 3.14.
Radius and Diameter of a Circle
=2
Ex 20
=
Circumference of a Circle
=2
=
Area of Circle
=
Find the radius of a circle if the diameter is 5.2 cm.
Page 10 of 14
FLC
Ch 10
Ex 21 Find the circumference of a circle with radius = 15 in. Provide both the exact answer and
estimate.
Ex 22 Jimmy’s truck has tires with a radius of 30 in. How many feet does his truck travel if the wheel
makes 9 revolutions?
Ex 23 Mickey’s car has tires with a radius of 15 in. He backed up his car a distance of 9891 in. How
many complete revolutions did the wheels make backing up?
Ex 24 Tom made a base for a circular patio by pouring concrete into a circular space 10 ft in diameter.
Find the cost at $18 per square yard.
Page 11 of 14
FLC
Ch 10
Sec 10.6: Volume
The volume of a cylinder is the area of its circular base ( r 2 ) times the height
(h). V  r 2 h , where r is the radius and h is the height.
The volume of a sphere is V 
The volume of a cone is V 
4r 3
where r is the radius.
3
r 2 h
3
The volume of a pyramid is V 
where r is the radius and h is the height.
Bh
, where B is the area of the base of the pyramid and h is the height.
3
Ex 25 Find each volume.
a) A cylinder with radius 3 m and height 8 m.
b) A sphere with radius 5 m.
c) A cylindrical trash can with radius 1.05 ft and height 3.6 ft.
d) A hemisphere with radius 6 m.
Page 12 of 14
FLC
Ch 10
e) A cone with height 12 ft and radius 6 ft.
f) A pyramid with height 10 m and a rectangular
base measuring 8 m by 14 m.
Ex 26 A collar of Styrofoam is made to insulate a pipe. Find the volume of the unshaded region (which
represents the collar). The large radius R is the outer rim. The small radius r is the edge of the
insulation.
=
=
=2
Sec 10.7: Similar Geometric Figures
Similar means that the things being compared are alike in shape, even though they may be different in
size.
Examples of Similar Triangles
Similar Triangles
The corresponding angles of similar triangles are equal. The lengths of corresponding sides of similar
triangles have the same ratio. The perimeters of similar triangles have the same ratios as the
corresponding sides. That is:
2
=
Page 13 of 14
FLC
Ch 10
Similar Figures
The corresponding sides of similar geometric figures have the same ratio.
Ex 27 Find the missing side. Provide the exact answer and also round to the nearest tenth when
necessary.
25 cm
8 cm
𝑛
3 in.
75 cm
15 in.
20 in.
𝑛
3 in.
Ex 28 Two triangles are similar. The larger triangle has sides 15 cm, 17 cm, and 24 cm. The 24-cm side
on the larger triangle corresponds to a side of 9 cm of the smaller triangle. What is the perimeter of the
smaller triangle?
Ex 29 Thomas is rock climbing in Utah. He is 6 ft tall and his shadow measures 8 ft long. The rock he
wants to climb casts a shadow of 610 ft. How tall is the rock he is about to climb?
Page 14 of 14