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Option #1: Counting Principles #1

Write a report that answers the following questions and meet the list of requirements that follows.

Be sure to describe or define the Fundamental Counting Principle, combinations, and permutations in your paper introduction. Include detailed calculations and solutions in the body of your paper.

Questions:

1. How many unique types of pizza can you create when allowed to choose two of seven types of meats, two of eight types of vegetables, one of two types of sauces, and one of three types of crusts?
2. A bowl of candy contains many pieces of candy but only six different types of candy. How many ways can three pieces of candy be removed from the bowl? Assume you can remove one or more pieces of the same type of candy.
3. A basketball team has sixteen women on the roster. In how many ways can the coach choose a roster consisting of five different positions?

Requirements:

• Your paper should be 2-3 pages in length and should cite and integrate at least one credible outside source.
• Include a title page, an introduction, a body, a conclusion, and a Reference list.
• The introduction should summarize the problem and state what approach and method will be applied to solve it.
• The body of your paper should answer the questions posed in the problem, explain how you approached and approached and answered the question or solved the problem, and for each question, show all steps involved.
• The conclusion should summarize your findings and what you have determined from the data and your analysis, with a broader or personal perspective in mind when applicable.
• As with all written assignments, provide in-text citations and a reference page.
• Include any tables of data or calculations, calculated values, and/or graphs associated with this problem in the body of your assignment.
• Document formatting, citations, and style should conform to the APA Requirements. In addition, the items noted in the ?Good Writing Tips? section should be observed. Refer to the document (attached at the bottom of the page) Calculating Tree Height as an example of how the paper is to be formatted.

Running head: CALCULATING TREE HEIGHT Calculating Tree Height

Craig Storlie

MTH109 ? Mathematical Explorations

Colorado State University ? Global Campus

Dr. Al Gebrawiz

September 10, 2013 1 CALCULATING TREE HEIGHT 2

Calculating Tree Height The height of tall objects can be difficult to measure using direct measurement. It may

also be dangerous and lack accuracy. An alternative to direct measurement is available by

incorporating trigonometry and other direct measurements. In this assignment, the height of a

large tree was calculated using the trigonometric tangent function.

Methodology and Discussion

In this problem, the height of a large tree is unknown. The tree is standing on horizontal

ground and is 150 feet from Point A. The angle between the ground and the tip of the tree is 40°,

and is measured at Point A (Figure 1). The goal of this problem is to determine the height of the

tree using trigonometry. Point A 40°

150 feet Figure 1. Dimensions used to calculate tree height.

One portion of trigonometry deals with right angle triangles ? those that contain a right,

or 90°, angle. The tangent function describes the tangent of an angle as equal to the length of the

side opposite of the angle divided by the length of the side adjacent to the angle (not the

hypotenuse). This relationship can be used to calculate the height of the tree shown in Figure 1.

tan 40° = 0.84= tree height

150 feet tree height

150 feet CALCULATING TREE HEIGHT 3 tree height =0.84 ( 150 feet )

tree height =126 feet Errors can arise from inaccurate estimates of the angular measurement. Table 1 shows the

errors that would occur if incorrect angular estimates were made in the problem solved in this

paper. Errors can be significant, and can be minimized through the careful use of a protractor or

other accurate sighting device.

Table 1

Results of Angular Measurement Errors in Calculating Tree Height

Angle

(degrees)

40 (actual)

45

50 Estimated Height

(feet) Error

(feet) Error

(%) 126

150

179 0

24

53 0

19

42 Note: Estimated Height is calculated using Estimated Height = tan ? (150 feet).

The method described in this paper provides an accurate estimate of object height if right

angle geometry exists and if the horizontal distance and angle measurements are accurately

Conclusions

Trigonometry is a powerful tool. In this assignment, the height of an object was

calculated given the distance an object was from a point and the angle measured between the top

of the object and the horizon. This method does not provide an accurate estimate if a right

triangle is not formed between the object and the horizontal plane, such as an object standing

vertically on sloped ground. However, other simple trigonometric methods exist that can be used

for more complicated geometries (Gebrawiz, 2013). CALCULATING TREE HEIGHT 4 CALCULATING TREE HEIGHT 5

References Gebrawiz, A. B. (2013). Trigonometric measurement methodology. Madison, WI: Bolt Upright

Publishing.

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This question was answered on: Oct 14, 2020

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