This is a diagnostic quiz, designed to challenge you and show us how you think.   You probably haven't been taught how to do most of the problems--we want to see if you can figure out how to solve them using what you know.  Many of these problems are very challenging, so don't be discouraged if you aren't sure of your work on a lot of them.  But give every one your best effort.  Use your common sense and what you know. 

This test doesn't count, but you will be given your results, so do your best.

Introduction to Part A

It will be very important in this course for your instructor to see and understand the process of visualization and reasoning you use when you solve problems. This exercise is designed to give you a first experience with these ideas, and your instructor a first look at your work.

Don't let this exercises prevent you from starting the pulse experiment. However

Answer the following questions and explain in commonsense terms why your answer makes sense.

For each question describe a picture you might draw to make sense out of the situation.

Samples

Here is a sample question and the general type of response we are looking for:

Sample question and response

If a bundle of shingles covers 30 square feet, how many bundles are required to cover a 600 square foot roof?

<Response cms00>

We might draw a picture of a rectangle representing the area, dividing the rectangle into a number of smaller rectangles each representing the area covered by a single bundle. This makes it clear that we are dividing the roof area into 1-bundle areas, and makes it clear why we are going to have to divide.

Reasoning this problem out in words, we can say that a single bundle would cover 30 square feet. Two bundles would cover 60 square feet. Three bundles would cover 90 square feet. We could continue in this manner until we reach 600 square feet. However, this would be cumbersome. It is more efficient to use the ideas of multiplication and division.

We imagine grouping the 600 square feet into 30 square foot patches. There will be 600 / 30 patches and each will require exactly one bundle. We therefore require 600 / 30 bundles = 20 bundles.

<End response>

Your responses might not be as clear as the above, though they might be even more clear. I won't be looking for perfection, though I wouldn't object to it, but for a first effort at visualizing a situation and communicating a reasoning process. This is not something you are used to doing and it might take a few attempts before you can achieve good results, but you will get better every time you try and you will eventually get the 'hang' of it.

You might be unsure of what to do on a specific question. In such a case specific questions and expressions of confusion are also acceptable responses. Such a response must include your attempts to come up with a picture and reason out an explanation. For example your response might be

Sample expression of confusion: I've drawn a picture of a pile of bundles and a roof but I'm not sure how to connect the two. I tried multiplying the number of bundles by the square feet of the roof but I got 18,000, and I know it won't take 18,000 bundles to cover the roof. How do you put the area covered by a bundle together with the roof area to get the number of bundles required?

A bad response would be something like "I don't know how to do #17". This response reveals nothing of your attempt to understand the question and the situation. Nor does it ask a specific question.

Incidentally, you might be tempted to quote rules or formulas about rates and velocities in answering these questions. Don't. This exercise isn't about being able to memorize rules and quote them. It is about expanding your ability to visualize, reason and communicate.

Part A

1.  If you earn 129.13 dollars in 10 hours, at what average rate are you earning money, in dollars per hour?

2.  If you are earning money at the average rate of 44 dollars per hour, how much do you earn in 8 hours?

3.  How long does it take to earn 114.6078 dollars at an average rate of 13 dollars per hour?

4.  If you travel 80.55376 miles in 10 hours, at what average rate are you traveling, in miles per hour?

5.  If a ball rolling down a grooved track travels 46.03593 centimeters in 8 seconds, at what average rate is the ball moving, in centimeters per second?

6.  If a ball travels at an average rate of 44 centimeters per second, how far does it travel in 7 seconds?

7.   How long does it take to travel 147.0698 miles at an average rate of 30 miles per hour?

Part B

Now solve the following by drawing a picture, and by constructing a graph.  You may round everything off to three significant figures.

1.  If your total earnings up to week 10 are 52.52809 dollars, and your total earnings up to week 17 are 58.52809 dollars, then what are your average earnings per week? Why do we say average earnings per week rather than just earnings per week?

2.  If you are earning money at an average rate of 7 dollars per week, and if at the end of week 10 your total earnings are 60.69826 dollars, then what will be your total earnings that the end of week 20?

3.  If your total earnings up to week 9 are 131.705 dollars, and if you earn money at a constant rate of 15 dollars per week, then at what week will your total earnings be 146.705 dollars?

4.  If an runner is at milepost 340.2328 after having traveled for 6 hours and at milepost 398.2328 after having traveled for 13 hours, then what is its average speed? Why do we say average speed rather than just speed?

5.  If the water in a uniform cylinder has depth 71.51243 cm at clock time 7 seconds and depth 83.51243 cm at clock time 18 seconds, then at what average rate is the depth changing? What we say average rate rather than just rate?

6.  If the depth of water in a uniform cylinder is changing at an average rate of 7 cm/second between clock times 6 s and 11 s, and if its depth at clock time 6 seconds is 35.35815 cm, then what is its depth at clock time 11 seconds?

Part C

A 7000 lb baby whale of a certain species is expected to gain weight at the rate of 40 pounds per day. If a whale at the age of 20 days weighs 6000 pounds, while at the age of 30 days the same whale weighs 7200 pounds, is is likely that this whale is of the specified species?

The rate at which a certain species reproduces in a certain ecosystem, when the ecosystem is healthy and when the population of the species is 130,000, should be 7000 per month. In the 16th month of recordkeeping the species population is 115,000, while in the 20th month the population is 130,000. What can be set about the health of the population and the ecosystem?

An artificial intelligence program is attempting to navigate a submarine through a mine field. The efficiency of its plan should change from near 0% to near 100% in the 12 milliseconds it has to avert disaster. The efficiency increases rapidly at first, and less rapidly as time goes on. If the program is functioning correctly, when the efficiency is 70% the rate at which efficiency improves should be approximately 12% per millisecond. After 3 milliseconds the efficiency is 50% and after 6 milliseconds the efficiency is 80%. Should we abandon ship?

Part D

1.  The logarithm of a number is the power to which 10 must be raised to get that number. Answer the following:

2.  Make a table for y vs. x if y = x^2, using x values -3, -2, -1, 0, 1, 2, 3. Then plot your points as ordered pairs on a y vs. x coordinate axis. Draw a smooth curve through your points.

wpe1.jpg (14798 bytes)

3.  A bunch of chemicals are mixed in a tank. The tank has a leak, and the rate at which the quantity y of chemicals available for reaction changes is related to the time on a timing clock by (rate of change of y) = (3 - t), where t is clock time in hours and the rate is in kg/hour. How many kg of chemicals do you think are lost during the second hour?

4.  The graph below depicts the unrestricted population growth of a certain species.

wpe5.jpg (12950 bytes)

5.  The temperature of a certain chemical reaction can be modeled by the function T = (t - 2.3)(t-5.7)(t-7.4), where T is the temperature in Celsius degrees and t is the clock time in minutes. At what clock times is the temperature 0?