Applied Optimization Lab

I’m studying for my Calculus class and don’t understand how to answer this. Can you help me study?

Directions:

Format: Use Geogebra to complete all problems. Solutions should be summited as a lab report. A “lab report” means that more explanation should be provided compared to a homework assignment. Your name should be located at the top of your report as well as Lab #6. Then, you should write the problem number and solution. Your solution should include all the work required to answer the question including explanations in complete sentences. You may choose to typeset or hand-write your solutions, but note that there are places where you will have to include graphics from Geogebra in your work. Problems should be in numerical order. Do not provide several pages of hand written work with all Geogebra graphics at the end. The report needs to be easy to follow.

Grading: This lab is worth 100 points. Details are provided below.

1a.) 5 pts 1b.) 15 pts 1c.) 5 pts 1d.) 10 pts 1e.) 10 pts 1f.) 5 pts

2a.) 5 pts 2b.) 15 pts 2c.) 5 pts 2d.) 10 pts 2e.) 10 pts 2f.) 5 pts

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Overview

This assignment explores two real world problems that require applied optimization to solve. (The necessary concepts are discussed in Section 3.4.). These problems involve functions that are relatively complicated, meaning that their derivatives will be challenging to work with. Instead of finding critical points by hand, you will learn how to use Geogebra to assist you in these algebraic and calculus-related computations.

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Problems:

1.) Two vertical poles of heights 60 ft and 80 ft stand on level ground, with their bases 100 ft apart. A cable is stretched from the top of one pole to some point on the ground between the poles, and then to the top of the other pole. You need to find the minimum length of cable that could be used to do this by following the steps below:

a.Draw a diagram illustrating this situation. Introduce notation/variables and label the diagram with your symbols.

b.Based on your diagram and labels in part (a), determine a formula that represents the total length of the cable needed as a function of the distance from the base of the one of poles to the point on the ground where the cable touches. Make sure to show your work and state the formula you determine. You MUST explain all calculations and equations used to create this formula. (You may want to check your answer with the professor before continuing.)

c.Find any restrictions on your independent variable. Write one sentence that discusses and justifies the domain of the function based on these restrictions.

d.What are the critical numbers of the function you found in part (b)? Use the Geogebra directions in the box below to complete this problem. Write your critical numbers as decimals rounded to 5 places and include units of measurement. State your final answer by writing a couple of sentences explaining what Geogebra did and referring to the screen shot of the work you completed in Geogebra. Note that I want a picture of your work in the input lines, not necessarily the graph. The graph will only show up if you change the numbers on the axes appropriately.

Geogebra Directions for Finding Critical Numbers

Step 1: Enter your expression from part (b) as a function, f(x), in Geogebra. This means, in the input line, we type the following and press enter.

f(x)=(put your function here)

Step 2: In another input line, type the following command and press enter.

Roots(f ’(x), 0, 100)

This single command takes the derivative of your function, and then solves the equation for where the derivative is zero on the interval [0,100]. (Root is another way of saying: the solution to an equation that is set equal to zero.) The command will report the roots as ordered pairs such as (3, 0). The first number in the ordered pair is the root (the number that causes the derivative to equal zero). The second number in the ordered pair tells us that the first derivative is zero at this point.

To change the number of decimal places, click on the Graphics View Toolbar on the upper right side of the screen. Then click on the gear icon . Finally click on the other gear icon that appears below the . Change the number of decimals as appropriate.

Include a screen shot of your Geogebra work for this part.

e.What is the minimum length of cable needed? Justify your answer using the calculus ideas we have been discussing before this. Explain what you are doing using complete sentences. Remember that you can have Geogebra calculate function values by typing f(a number) into an input line.

f.Where should we locate the place where the cable touches the ground in order to achieve the minimum length? Be specific.

2.) A pipeline for transporting oil will connect two points A and B that are 3 miles apart (as the crow flies) and on opposite banks of a straight river one mile wide. (Note that A and B must not be directly across from each other based on the information given so far.) Part of the pipeline will run under water from point A to a point C on the other side of the river. Then the pipeline will run above ground from point C to point B. If the cost per mile of running the pipeline under water is three million dollars a mile, while the cost per mile of running it above ground is two million dollars a mile, find the location of point C that will minimize the cost (ignoring the slope of the river bed). Hint: Let D be the point directly across the river from A so that C is between D and B.

a.Draw a diagram illustrating this situation. Introduce notation/variables and label the diagram with your symbols.

b.Based on your diagram and labels in part (a), determine a formula that represents the total cost of the pipeline. Make sure to show your work and state the formula you determine. You MUST explain all calculations and equations used to create this formula. (You may want to check your answer with the professor before continuing.)

c.Find any restrictions on your independent variable. Write one sentence that discusses and justifies the domain of the function based on these restrictions.

d.What are the critical numbers of the function you found in part (b)? Use the Geogebra directions in the previous problem to do this. Write your critical numbers as decimals rounded to 5 places and include units of measurement. State your final answer by writing a couple of sentences explaining what Geogebra did and referring to the screen shot of the work you completed in Geogebra.

e.What is the minimum cost of the pipeline? Justify your answer using the calculus ideas we have been discussing before this. Explain what you are doing using complete sentences. Remember that you can have Geogebra calculate function values by typing f(a number) into an input line.

f.Where should we locate point C in order to minimize the cost of the pipeline? Be specific.

NOTE: The first document is the textbook. The second document is the previous assignment to use as guideline for this assignment.

Geogebra Website:

https://www.geogebra.org/classic


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Assignment Solutions


Assignment Solutions

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