Geometry Lab

In this lab you will determine the cost of putting a roof on a barn. In order to find a true cost you will need to know the exact area of the roof, the cost of materials, and the cost of labor.

The barn is in need of a total roof replacement. The “Farmer” has already taken the roof off and wishes to put a new tin roof on. To put on a tin roof all you need to do is lay down sheets of plywood and nail the tin on top of the plywood.

Find out how much it would cost a roofer to cover the roof with plywood and then nail tin to the top of the plywood.

Part I. Finding the area of the roof

In order to determine the amount of plywood and tin that is needed for the roof you have to determine the area of the roof. Use the blueprints below to determine the total amount of area the roof of the barn will cover. (For accuracy purposes assume that the overhang is calculated in the distance given)

Notice that the barn is symmetrical. On each side of the barn we have “roof A” and “roof B” which are both rectangular in shape and have a length of 30 ft.

Probability Assignment:

The two-way table below summarizes how students performed in three sections of MAT 152 on the Unit 1 Test. The row labels are letter grades. The column labels describe whether or not a student completed both chapter quizzes for the unit (first column), or missed at least one quiz for the unit (second column).

Completed both quizzes / Missed at least one quiz / Total
A / 28 / 1 / 29
B / 11 / 3 / 14
C / 13 / 1 / 14
D/F / 3 / 11 / 14
Total / 55 / 16 / 71
  1. What is the probability that a randomly selected student earned an A on the test?
  2. What is the probability that a randomly selected student completed both quizzes?
  3. What is the probability that a randomly selected student earned a C or better on the test?
  4. What is the probability that a student who misses at least one quiz will earn an A on the test?
  5. What is the probability that a student who completes both quizzes will earn a D or F on the test?
  6. What is the probability that a randomly selected student earned a D/F AND completed both quizzes?
  7. If two students are randomly selected without replacement, what is the probability that both earned a C?
  8. Are the events “Earned a C or better on the test” and “Completed both quizzes” independent or dependent? You must justify your answer mathematically.
  9. Are the events “Earned an A on the test” and “Missed at least one quiz” mutually exclusive? Justify your answer using the table.
  10. What is the probability that a randomly selected student earned a D/F OR missed at least one quiz?

Budget Lab:

Introduction:

Bungo and Esmerelda Baggins and their 13 year old son, Fosco, live in a beautiful house in Cary, North Carolina. Their house has 4 bedrooms, 4 baths and is 3,475 ft2. In 2012, shortly after the family purchased their home and a new Nissan Altima, Bungo lost his job. Since then he has been unable to find employment. Living on Bungo's unemployment insurance rather than his salary left the family with less money than they were used to spending, so they used credit cards to pay for some of their expenditures. In December of 2013, believing that he was about to be offered a prestigious new job, Bungo leased a 2014 Honda CR-V (36 month lease). That job fell through and his unemployment benefits have recently run out. The family can no longer meet their monthly expenses and have come to your group for help.

Directions:

Read the information found at:

and watch the videos found at

(hint: if you have trouble watching some of the videos, read the transcript).

Look at the Baggins family expenses found on the attached Excel sheet. Research ways in which the family can cut its monthly expenses and balance its budget. Your recommendations must be realistic (for example the Baggins family can't stop eating or wearing clothes) and well researched. You cannot totally eliminate any expense, only find ways to reduce it. If you recommend saving money on housing, insurance, car payments, utilities, etc. you must research that budget item and find a less expensive alternative for them. If you recommend debt consolidation, you must find them a lower cost loan. Divide the research up among your group then consolidate your recommendations on your Final Worksheet. Fill out the New Budget column on the Baggins_Budget_Worksheet with your recommended changes. Be sure that their new budget covers all of their monthly expenses. Document your recommendations and cite your research in the text box on the worksheet.

Normal Distribution Lab:

Answer the questions below. When necessary, show the Excel functions, along with inputs, used to obtain your answers.

  1. Is a normally distributed random variable quantitative or qualitative?
  2. Complete the statement: The area under a Normal curve represents …
  3. Complete the statement: The values on the x-axis of a Normal curve represent …
  4. Explain in the space below how the Normal distribution and Standard Normal distribution are related (discuss shape, parameters, probability). Be specific and use complete sentences.
  5. The time taken to assemble a car in a certain plant is a random variable that follows a normal distribution with a mean of 20 hours and a standard deviation of 2 hours.
  6. What is the probability that the time it takes to assemble a car is between 22 and 24 hours?
  7. Find probability that a car can be assembled in less than 17 hours.
  8. Is it unusual to assemble a car in less than 17 hours? Use your answer to part (b) to explain.
  9. A radar unit is used to measure speeds of cars on a motorway. The speeds are normally distributed with a mean of 90 km/hr and a standard deviation of 10 km/hr.
  10. The fastest 5% of drivers may be considered to be driving recklessly. At what speed would a car be considered reckless?
  11. The middle 90% of cars are driving between which two speeds?
  12. Does a sampling distribution (its shape, parameter values, probabilities) depend on the size of the samples? Use complete sentences to explain why or why not.
  13. We know that the standard deviation of a sampling distribution is smaller than that of its parent population. Explain in the space below why this occurs (do not say it is because the standard deviation is divided by the square root of the sample size – I’m not asking for the formula, but why this formula works). Use complete sentences.
  14. The amount of saturated fat in a daily serving of a particular brand of breakfast cereal is normally distributed with mean 25 g and standard deviation 4 g.
  15. Find the sampling distribution of the daily average saturated fat intake over a 30-day period(one month). Include the mean and standard deviation in your answer, as well as the name of the distribution.
  16. What is the probability that the average daily saturated fat intake for the month was more than 27 g?

Conic Sections in the Real World

Conic sections occur in natural, mechanical, and physical settings in the world around us – the orbits of the planets around the sun are elliptical, the path of a projectile (such as a cannonball) is parabolic, and the patterns formed on a wall by a lamp shade are hyperbolic. While many of these phenomena would be difficult to measure, your goal in this project is to find conic sections that exist in your everyday lives and analyze them further. Research the topic on the internet to get more ideas of how/where conics exist in nature.

For this project, you are going to submit a technology based final product. You may create an animated powerpoint slide show, a movie, or any other type of ‘non-paper’ presentation that you choose. Your presentation must include visuals, and may also include audio. Whatever format you choose should be viewable in 2-3 minutes and should get the ideas described below across in an easily understandable fashion. Talk to me to confirm the presentation format you choose if it’s not one of the ones listed above.

Requirements:

1. Take pictures or videos of the conics you discover in the world around you – these must be your images, not pictures taken from the internet. Show as many of these as you would like to in your presentation, with a minimum of 5 different images – look for things that are especially different and unique, be creative!

2. Choose 2 different conics from the ones you discovered that you can physically measure, then find the equation of each conic showing your work clearly in your project. Be sure to show/include measurements on the pictures of the conics chosen – lay a ruler across the conic when you take its picture.

3.Use dialogue (either verbal or written) throughout the presentation. Make sure that your visuals and explanations are clear and correct.

4.Discuss the connection of the mathematics you discovered to the real world. Explain how this process helped you to better understand conic sections.

Timber Lab

North Carolina produces a lot of lumber each year from its public and private forest land. Suppose you were a tree farmer and you wanted to know just how much lumber you could harvest from your land. The hard way to do this is to cut down all the pine trees, take them to a mill and see how much lumber you obtain. This could be a minor problem if you didn’t own the land and were just trying to estimate the value of the land. An easier, and of course mathematical, way of doing this would be to mark off a small piece of the land in question and use this small sample to represent the whole forest. It won’t be completely accurate, but if you are careful you should be able to get a good estimate. (This is how they do it in the real world.)

To mimic this estimation process, you will be assigned a triangular plot of land with pine trees on it. By determining the size and number of pines in the triangular region, and extrapolating, you should be able to predict how many of each size and type of pine tree would be in a larger stretch of forest. Knowing the volume of each size and type of pine tree you had in your region, you would then know the volume of lumber for a similar tract of forest land.

Below is data taken from a stand of pines on Wake Tech land. Your instructor will assign a table of data to each group. Each group will only work with one set of data.

Data Set A. 11 Tree Triangle

Triangle Sides 73.4ft, 63.4 ft, 35.5 ftTree Type / Number of Trees / Circumference of Representative Tree (in) / Angle of Elevation (degrees) / Angle of Depression (degrees) / Distance to Tree (ft)
Small / 6 / 25.0 / 43 / 3 / 46
Medium / 2 / 39.5 / 32 / 2 / 85
Large / 3 / 58.8 / 37 / 6 / 72

Part 1: Calculation Guidelines

A. Determining the area of your region.

Determine the area of your region. The triangular region is not a right triangle. Include a drawing of your region with measurements clearly labeled and show all calculations used to find the area.

B. Determine the radius of the trees in your region.

Determine the radius of the base of your representative tree for each category. Show all calculations.

C. Determine the height of your representative trees using trigonometry.

Assume the tree is perpendicular to the ground. Use the distance and angles given and right triangle trigonometry to calculate the heights of each representative tree. Draw and label diagrams and show all calculations used.

D. Determining the volume of a tree.

You will need to calculate the volume of your representative trees. Show all calculations used. Hint: A pine tree can be treated as a right circular cone.

E. Determining the total volume of the trees.

Using the volumes of the representative trees and the number of each type of tree, calculate the total volume of the trees in your region. Show all calculations.

Part 2: Analysis

Do you think the calculations you made above accurately represent the amount of lumber on the land? Why or why not? Explain what factors you might have missed and where error might have occurred. How would you improve this procedure?

Part 3: Extrapolation

If you had a similar tract of land to the one you measured, but it had an area of 23 acres, what would be the volume of the pine trees on the land? Show all calculations.

Part 4: Presentation

Each group will hand in only one project write up. It should be neat, in order and look as if it was completed by one person.