Download (bring lecture 3 notes to complete the discussion of area, perimeter

Math 117 Lecture 5 notes
(bring lecture 3 notes to complete the discussion of area, perimeter, circumference, and arcs)
Problem: Larry purchased a plot of land surrounded by a fence. The former owner had subdivided
the land into 13 equal-sized square plots, as shown. To reapportion the property into two plots of
equal area, Larry wishes to build a single, straight fence beginning at the far left corner (point P on
the drawing). Is such a fence possible? If so, where should the other end be?
For more area problems, see figure 11-24 in Billstein.
Area problems:
1. Use a diagram to explain the formula for the area of a parallelogram using the area of a
rectangle.
2. Use a diagram to explain the formula for the area of a triangle using the area of a
parallelogram.
3. Use a diagram to explain the formula for the area of a trapezoid using the area of a rectangle.
Hero or Heron’s Formula for area of any triangle:
5
Area =
√s(s-a)(s-b)(s-c)
4
7
where s = semi-perimeter = 1/2(a+b+c)
Angle Measurement
An angle is measured according to the amount of "opening" between its sides. The degree is
commonly used to measure angles.
A complete rotation about a point has a measure of 360°. A degree is subdivided into 60 parts,
called minutes, and each minute is further subdivided into 60 parts, called seconds. The
measurement 29 degrees, 47 minutes, 13 seconds is written 29°47'13".
Problem:
Find 47°45' – 29°58'
Express 47°45' as a decimal number of degrees
Express 32.6° in degrees and minutes, & seconds, without a decimal
Triangulation:
Forest rangers use degree measures to identify direction and locate critical spots
Such as fires. Suppose a forest ranger at tower A observed smoke at a bearing of
149° clockwise from North, while another ranger at tower B observes the same
source of smoke at a bearing of 250° clockwise from North. Explain how the forest
rangers could find the location of the fire.
A 149°
B
Angles can also be measured in radians.
One radian is the measure of an angle in standard position whose terminal side intercepts an arc of
length r (radius).
r
r
1 radian
Because the circumference of a circle is 2πr, there are 2π radians in a full circle. Degree measure
and radian measure are therefore related by the equation: 360° = 2π radians, or 180° = π radians.
You can use these equations to convert degrees to radians and radians to degrees.
a. Try converting 30° to radians. b. Try converting π/6 radians to degrees.
Linear Notions: (see page 464 Table 9-2)
Collinear points
A point between two points on a line
Line segment
Ray
Planar Notions:
Coplanar
Skew lines
Intersecting lines
Concurrent lines
Parallel lines
Question: Why is a tripod considered “level?”
Properties of Points, Lines, and Planes
1. There is exactly one line that contains any two distinct points.
2. If two points lie in a plane, then the line containing the points lies in the plane.
3. If two distinct planes intersect, then their intersection is a line.
4. There is exactly one plane that contains any three distinct noncollinear points.
5. A line and a point not on the line determine a plane.
6. Two parallel lines determine a plane.
7. Two intersecting lines determine a plane.
When a transversal intersects two parallel lines, many types of angles are formed: exterior,
interior, vertical, corresponding, supplementary.
When a transversal intersects two parallel lines, the alternate interior angles are congruent.
When a transversal intersects two parallel lines, the corresponding angles are congruent.
Any point P on an angle bisector is equidistant from the sides of the angle.
Any point that is equidistant from the sides of an angle is on the angle bisector of the angle.
Any point equidistant from the endpoints of a segment is on the perpendicular bisector of the
segment.
Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the
segment.
The following hold for every isosceles triangle.
The angles opposite the congruent sides are congruent.
The angle bisector of an angle formed by two congruent sides is an altitude of the triangle as well
as the perpendicular bisector of the third side.
Suppose we have 3 points, A, B, C, in a plane and can accurately measure the distance between any
two of them. How can we determine if the points are collinear?
1. find length of the line segments AB, BC, and AC
1. If the points are not collinear, the sum of any two lengths will be greater than the third. If the
points are collinear, the sum of two of the line segments is equal to the third.
Example:
A
B
C
AB + BC = AC
But what if AB + BC > AC ?
Triangle Inequality Property: The sum of the measures of any two sides of a triangle must be
greater than the measure of the third side.
If two sides of a triangle are 31 cm and 85 cm long and the measure of the third side must be a
whole number of centimeters,
2. what is the longest the third side can be?
3. What is the shortest the third side can be?
Geometric solids
1. The lateral surface area of a solid is the sum of the areas of the sides excluding the area of
the bases.
2. The total surface area of a solid is the sum of the lateral surface area plus the area of the
bases.
3. The volume of a solid is the number of cubic units of measure contained in the solid.
Prism
Pyramid
Cylinder
Cone
Sphere
See table 9-10 for more information on semiregular polyhedra.
Volumes:
Bh
1/3 Bh
Bh
1/3 Bh
4/3 πr3
Painting houses, buying roofing, seal-coating driveways, and buying carpet are among the common
applications that involve computing areas. In many real-world problems, we must find the surface
areas of such three-dimensional figures as prisms, cylinders, pyramids, cones, and spheres.
Formulas for finding these areas are usually based on finding the area of two-dimensional pieces of
the three-dimensional figures. We will use the notion of a net, a two-dimensional pattern that can
be used to construct three-dimensional figures, to aid in determining surface areas of the figures.
net
Surface Area of Prism: To find the surface area of a right prism, we find the sum of the areas of
the rectangles that make up the lateral faces and the areas of the top and bottom.
Surface Area of a Pyramid: A right regular pyramid is a pyramid such that the segments
connecting the apex to each vertex of the base are congruent and the base is a regular polygon.
The lateral bases of the right regular pyramid are congruent triangles. Adding the lateral surface
area n(1/2 bl) to the area of the base B gives the surface area.
Surface Area of a Cylinder: To find the surface area of a right circular cylinder, we cut off the
bases and slice the lateral surface open by cutting along any line perpendicular to the bases. Then
we unroll the cylinder to form a rectangle. Tot total surface area is the sum of the area of the
rectangle and the areas of the top and bottom circles.
Surface Area of a Cone: It is possible to find a formula for the surface area of a cone by
2
approximating the cone with a pyramid. Thus, surface area = πr + πrl.
Surface Area of a Sphere: The surface area of a sphere is simple using calculus, but not
2
elementary mathematics. Surface area = 4πr
Real-Life application of surface area: One way to calculate patient dosage is by finding the
patient’s body surface area (BSA) using a nomogram. To use a nomogram you must know the
patient’s height and weight. Dosage calculations using BSA are considered by some to be more
accurate than body weight calculations, and are typically used to calculate pediatric dosages.
2
Ex: ordered: Adriamycin PFS 2 mg/m
The child is 48 in. tall and weighs 50 lbs.
Volumes of Geometric Solids:
Volume of right rectangular prisms can be determined by measuring
how many cubes are needed to build it.
V = lwh
The volume of a cylinder is the product of the area of the base B and the height h.
2
V = Bh = πr h
Volumes of Pyramids and Cones: Students should explore the relationship by filling the pyramid
with water, sand, or rice and pouring the contents into the prism. They will find that the volume of
a pyramid is equal to one-third the volume of the prism. The same holds true for the relationship
between a cone and a cylinder.
V = (1/3) Bh
or
V = (1/3)πr 2h
Volume of a sphere: Imagine that a sphere is composed of a great number of congruent pyramids
with apexes at the center of the sphere and that the vertices of the base touch the sphere. If the
pyramids have very small bases, then the height of each pyramid is nearly the radius r. Hence, the
volume of each pyramid is (1/3)Bh or (1/3)Br. If there are n pyramids each with base area B, then
the total volume of the pyramids is V = (1/3)nBr. Because nB is the total surface area of all the
bases of the pyramids and because the sum of the areas of all the bases of the pyramids is very
close to the surface area of the sphere, the volume of the sphere is given by
2
3
V = (1/3)(4πr )r = (4/3) πr
Cavaleri’s Principle: Two solids each with a base in the same plane have equal volumes if every plane
parallel to the bases intersects the solids in cross-sections of equal area. (see page 649 in text)
Problem 1: If each edge of a cube is increased by 30%, by what percent does the volume increase?
Problem 2: A tennis ball can in the shape of a cylinder holds three tennis balls snugly. If the radius
of a tennis ball is 3.5 cm, what percentage of the tennis ball can is occupied by air?
- Handout problems Trigonometry is the study of right triangles.
Consider a right triangle, and choose one angle as the point of reference.
hypotenuse
opposite
q
adjacent
Ratios of the right triangle’s three sides are used to define the six trigonometric ratios:
sin q =opp/hyp
cos q =adj/hyp
tan q = opp/adj
csc q =hyp/opp
sec q =hyp/adj
cot q = opp/adj
Two special right triangles occur frequently in trigonometry. The ratios of their sides should be
learned.
45°
30°
√2
√3
1
90
1
°
45
°
90°
1
2
60°
Trig can be used to find the measure of a missing side of a right triangle.
Note that trig functions seem to go in cycles:
sin 0° = 0
sin 30° = 1/2
sin 120° = √3/2
sin 210° = –1/2
sin 300° = –√3/2
sin 45° = √2/2 sin 135° = √2/2
sin 225° = – √2/2 sin 315° = –√2/2
sin 60° =√3/2 sin 150° = 1/2
sin 240° = –√3/2 sin 330° = –1/2
sin 90° = 1
sin 180° = 0
sin 270° = –1
sin 360° = 0
The values of trig functions can also be found using a calculator.
Ex 1: Find the area of the right triangle.
6
20°
Ex 2: Find the area of the right triangle.
5
30°
Lab #5: Volumes lab using geo-solids. Note: the material in this lab is included on Unit 1 Exam.