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MATH 20063 CHAPTER 10 EXAMPLES AND DEFINITIONS Section 10.1 Systems of Measurement It can be helpful to have an everyday approximate equivalent to units of measure. These are called benchmarks. Some benchmarks for customary units: •1 inch – diameter of a quarter, length of a small paperclip •2 inches – width of a credit card •6 inches – length of a dollar bill •1 foot – length from knuckles of a hand to one’s elbow

•1 yard – height from floor to doorknob Volume is the amount of space occupied by an object. Capacity is the amount that can be contained by an object.

•Liquid volume: ounce, cup, pint, quart, gallon, barrel, hogshead •Dry volume: pint, quart, gallon, peck, bushel

Mass is a measure of the quantity of matter. Weight is a measure of how heavy something is. Weight is caused by the force of gravity pulling down upon an object. An object's weight depends on what planet or moon it's on (unlike mass, which is constant). •Ounce, pound, stone, hundredweight, ton Ex. A) Using appropriate units – which is the best unit to describe

• the amount of water in a swimming pool? (a) cup (b) pint (c) quart (d) gallon

• a bicycle's length? (a) inches (b) feet (c) miles (d) yards

• weight of a whale? (a) ounces (b) pounds (c) tons (d) too heavy to weigh

• weight of a pencil? (a) ounces (b) pounds (c) tons (d) too light to weigh

• can of soda (liquid measure)? (a) pints (b) quarts (c) gallons (d) ounces

Ex. B) The distance around a running track is 50π + 360 yards. (a) What is a more useful way to express this distance? (b) Is this value exact or approximate?

CUSTOMARY/METRIC CONVERSIONS KNOW: 1 inch ≈ 2.54 centimeters 1 mile ≈ 1.6 kilometers 1 quart ≈ 1 Liter 2.2 pounds ≈ 1 kilogram

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A few mnemonics for metric prefixes:

kilo hecto deka unit deci centi milli

Kind Happy Doctors Usually Do Care More

kilo hecto deka meter/liter/gram deci centi milli

Kathy Hall Drinks Milk/Lemonade/Gatorade During Class Mondays

King Henry Danced Merrily/Lazily/Gleefully Down Center Main

Incidental Note: The abbreviation ‘K’ is now commonly used for 1000 of something – dollars, meters, or bytes, for example. An actual kilobyte is 1024 bytes; the prefix ‘kilo’ is a convenient approximation. One megabyte is one kilobyte squared; one gigabyte is one kilobyte cubed. Some metric benchmarks: for each of the following metric units (in order to become comfortable with each size):

•kilometer – 9 football fields •meter – doorknob to floor •centimeter – width of index fingernail •millimeter – thickness of a dime

Ex. C) Use dimensional analysis to convert each of the following:

(a) 6 mm = cm (b) 125 cm = m (c) 5 inches = cm Ex. D) Choose an appropriate unit of measure for each measurement

(a) distance from home to a grocery store (b) weight of a refrigerator (c) water in a fish tank (d) height of a door (e) perfume in a bottle of perfume

Ex. E) My Celsius thermometer says I have a fever of 40°C. Should I be alarmed? The greatest possible error (GPE) of a measurement is half of the smallest unit used in the measurement. Ex. F) (a) If I give my height as 5 feet, 5 inches, what is the GPE? What heights might I be between? (b) Convert 5 feet, 5 inches to centimeters. What is the GPE of this measurement? Consider what GPE might or might not mean when using converted values.

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Section 10.2 Perimeter and Area The perimeter of a simple closed curve is the distance around the curve. THE PYTHAGOREAN THEOREM: If a right triangle has legs of lengths a and b and a hypotenuse of length c, then c2 = a2 + b2. Ex. G) Find the perimeter of the following figure:

• • • • •

• • • • •

• • • • •

• • • • •

The circumference of a circle is its perimeter. The ratio of the circumference of a circle to its diameter is always pi. Ex. H) If a bicycle wheel has a radius of 20 inches, how many times does it turn in going 1000 feet? Area is the measure of a region – i.e. the measure of the interior of a closed curve. Ex. I) What is the area of a third of a square with side length 23.7 cm? Ex. J) Find the area of the figure shown in example G above.

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Ex. K) True/False – counterexample: (a) Two figures that are congruent will have the same perimeter. (b) Two figures that have the same perimeter are congruent. (c) Two figures that are congruent will have the same area. (d) Two figures that have the same area are congruent. BONUS: A drawbridge measures 136 feet across when closed. Each side of the bridge opens to a height of 60 feet to allow boats to pass through. How wide is the opening when the drawbridge is open? Section 10.3 Areas of Quadrilaterals, Triangles, and Circles Part 1: Polygonal Regions The altitude or height of a triangle is the perpendicular segment from any vertex to the line that contains the opposite side. The median of a triangle is the segment from any vertex to the midpoint of the opposite side. Area Formulas For Polygonal Regions – Know and be able to Develop 1. The area A of a rectangle that has length l and width w:

A = lw. 2. The area A of a triangle that has base b and height h:

A = ½bh. 3. The area A of a parallelogram that has base b and height h:

A = bh. 4. The area A of a trapezoid that has parallel sides of lengths b and c and height h:

A = ½( b + c)h. The quantity ½( b + c) is referred to as the average base. Ex. L) An equilateral triangle has side length of s. (a) Find the height of the triangle. (b) Find the area of the triangle.

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Ex. M) This figure has a total height of h, an upper width of w and a lower width of ¾ w. Find an expression for the area of the region.

Ex. N) The length of a side of square ABCD is 1 and E is a point on side AD. If the area of triangle ECD equals 1/3 of the area of square ABCD, what is the length of segment ED? One more area formula: The area of a kite is ½ the product of the diagonals. Apothem – the distance of a perpendicular segment from the center of a polygon to a side of the polygon. Ex. O) Suppose a regular octagon has side length of 4 and the apothem has length 2 3 . Find the area of the octagon.

Practice Problems for Section 10.3, Part 1 – Polygonal Regions 1. The length of a rectangle is 4 yards longer than its width. If the perimeter of the rectangle is 48 yards, find the area of the rectangle. 2. The area of a parallelogram is 216 square inches, and its height is 9 inches. Find the base of the parallelogram. 3. The width of a rectangle is 2x and its length is y. If the area is 96 square units, express x in terms of y. 4. The bases of a trapezoid are x + 1 units and 2x – 2 units; its height is 3 units. If the area of the trapezoid is 21 square units, find x. 5. Suppose that the diagonals of a kite measure 100 millimeters and 200 millimeters. (a) Find the perimeter of the kite. (b) Find the area of the kite. [Problems 3 – 5 are variations of TAKS prep questions for high school geometry.]

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Part 2: Circles Ex. P) The radius of the larger circle below is 12 units. The intersection point of the two smaller circles is the center of the larger circle. (a) Find the combined area of the two smaller circles in the figure below. (b) Find that part of the area of the outer circle not included within the two smaller circles.

Radian – a unit of angular measure. 2π radians make up a circle. Since the circumference of a circle is 2π times the radius of the circle, this means that the length of the radius of a circle is equal to the arc length corresponding to an angle measuring one radian.

Ex. Q) Complete the table below:

Degrees Radians 360 180 90 60 45 30

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Ex. R) (a) Convert 4.5π radians into degrees. (b) Convert 165° into radians. (c) Approximately how many degrees are in 1 radian? Mark off a central angle within a circle. Then the subset of the circle bounded by this angle is an arc of the circle; the area of the wedge is a sector of the circle.

Ex. S) A circle has a radius of 4 cm, and an arc length of 5 cm. What is the measure of the central angle? Give your answer in radians. Ex. T) (a) Find the area of a sector whose central angle has a measure of π/3 and whose radius is 12 cm. (b) Find the area of a quarter circle with radius of 10 cm.

Practice Problems for Section 10.3, Part 2 – Circles 6. The diameter of the smaller circle is 6 units. The diameter of the larger circle is 10 units. Find the area of the outer ring in the figure below. Give your answer as an exact value.

7. Find the area of the shaded region in the figure above to the right. Give your answer as an exact value.

>> continued <<

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Practice Problems for Section 10.3 Part 2, continued 8. Fill in the following blanks: (a) 60° = radians. (b) 240° = radians.

(c)

!

"

6 radians = ° (d) 4 radians = °

9. For each of the following circles, find the missing arc length, a, radius r or angle measure, θ. Then find the area of each sector shown.

BONUS: In the figure below, the two circles with centers O and P each have a radius of 6 inches. How long is the darkened portion of the circumference of these circles, in inches?

[Problems 7 & 9 adapted from Modern Mathematics, 11th edition, Wheeler & Wheeler.]

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Section 10.4 The Pythagorean Theorem CONVERSE OF THE PYTHAGOREAN THEOREM: If the square of the measure of one side of a triangle is equal to the sum of the squares of the measures of the other two sides, the triangle is a right triangle. THE TRIANGLE INEQUALITY: The sum of the measures of any two sides of a triangle is greater than the measure of the third side. Ex. U) Consider the lengths 3, 4, and c, where c > 4. For what value(s) of c could these lengths form (a) some triangle? (b) a right triangle? (c) an acute triangle? (d) an obtuse triangle? Ex. V) Two bugs are in the top corner of a room that is 12 feet wide by 16 feet long by 8 feet high. On the floor in the extreme opposite corner is a piece of candy. What is the shortest distance to the candy for a bug that (a) can fly? (b) must crawl?

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Sections 10.5/10.6 Surface Area/Volume The total surface area of a closed space figure is the sum of the areas of all of its surfaces. Volume is the amount of space occupied by a closed space figure. I. Prisms and Cylinders Ex. W) I want to cover a cylindrical cat tree with carpet, except for its bottom. The cat tree will be 3 feet tall, and have a diameter of 26 inches. How much carpet do I need?

Ex. X) A container in the shape of a right rectangular prism is 110 centimeters high and contains 49.5 liters. Find the area of the base of the prism.

Ex. Y) If I take a piece of 8½” by 11” paper, there are two ways in which I can turn it into a cylinder. If I were to add a base to each, would one cylinder hold more popcorn than the other? If so, which?

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II. Pyramids and Cones A slant height is the height of one of the lateral sides of a pyramid.

SURFACE AREA OF A PYRAMID:

!

S.A. =1

2Pt + B

Ex. Z) Find the surface area of a right pyramid with square base with a side length of 30 cm and an altitude of 20 cm.

VOLUME OF A PYRAMID:

!

V = 13Bh

Ex. AA) I want to fill a pyramid with sand. The pyramid has a square base that is 42 in. long on each side, and has a slant height of 35 in. How much sand do I need? SURFACE AREA OF A CONE:

!

S.A. = "rt + "r 2 VOLUME OF A CONE:

!

V = 13Bh

III. Spheres SURFACE AREA OF A SPHERE:

!

S.A. = 4"r 2 VOLUME OF A SPHERE:

!

V = 43"r3

Ex. AB) Suppose that the radius of a sphere is doubled. (a) What happens to its surface area? (b) What happens to its volume?

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Ex. AC) (a) A cylindrical log with a circular base is to be sawn into a square column. Find the volume of the wood that will be wasted in terms of the length and radius of the log, assuming that the radius of the log goes from the center to a vertex of the square base. (b) Find the volume of wood that will be wasted for an 8 foot log with a circumference of 56.5 inches, using 3.14 as an approximation of pi. Give your answer to the nearest whole cubic foot.

Practice Problems for Sections 10.5/10.6 1. One cubic centimeter of iron has a mass of about 8 grams. What is the mass of an iron bar that measures 50 cm by 20 cm by 5 cm? 2. A cylindrical tank has a diameter of 10 feet. As the tank is being filled, how many cubic feet of water are needed to raise the water level in the tank by 5 inches? Give your answer as an exact value. 3. Find the volume and surface area of the figure below.

Practice problems 2 & 3 adapted from Modern Mathematics, 11th edition, Wheeler & Wheeler. BONUS: [2 points] A 10 x 10 x 10 cube made up of smaller 1 x 1 x 1 cubes is painted red. How many of the smaller cubes have paint on them? Show all work or an explanation of how you reached your answer.

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GIVEN FORMULAS FOR CHAPTER 10 TEMPERATURE CONVERSION

C° = 5

9 (F° ! 32)

SURFACE AREA OF A PYRAMID

S.A. = 1

2Pt +B

VOLUME OF A PYRAMID

!

V = 13Bh

SURFACE AREA OF A CONE S.A. = ! r t+ ! r 2 VOLUME OF A CONE

!

V = 13Bh

SURFACE AREA OF A SPHERE

S.A. = 4! r 2 VOLUME OF A SPHERE

!

V = 43"r3