MATH 117: Activity 1 Reader: Elizabeth
Section 1.1: Function Representations Recorder: Blake Munro
February 4, 2015 Researcher: Stephanie
Warning: You will need your results from this for a future worksheet, so be sure to save them!
Crickets are one of nature’s more interesting insects, partly because of their musical ability. In England, the chirping or singing of a cricket was once considered to be a sign of good luck. In China and Japan, they were kept in fancy cages in the house so the residents could enjoy their singing. Many of us are so used to hearing this sound on a summer evening that we would probably think something was wrong if it was missing. The male cricket “sings” to attract female crickets—not just to keep you up at night—and does this by rubbing his two front wings together.
1. Words: One of the interesting facts about crickets is that their activity is dependent upon the temperature. As a result of this, they can be thought of as ‘natural’ thermometers. The rate of a cricket’s chirp increases as the temperature increases and depends on the type of cricket. So if you know the right formula and the type of cricket, when you hear chirping, you can estimate the temperature by counting the chirps. Changes in humidity and different crickets of the same type also produce variations in a cricket’s chirping rate. The dominant factor, however, is temperature, so formulas relating temperature to the number of chirps are fairly accurate. Chirp rate will ultimately be measured in number of chirps per minute. Below are the rules for finding the temperatures, in degrees Fahrenheit, for three different types of crickets:
· The field cricket is the black cricket that is commonly found in the United States. For a field cricket, you need to count the number of chirps in fifteen seconds and add 38 to obtain the temperature.
· The tree cricket is small, pale green and usually found on trees. For the cricket commonly called the tree cricket, the temperature can be obtained by counting the number of chirps in seven seconds and adding 46.
· The snowy tree cricket is the species whose music is most in tune with that of the temperature since it is believed to be the most accurate. For this cricket, you need to count the number of chirps in fourteen seconds and add 42.
a. Let T represent that temperature and n the chirp rate. What are letters which representing quantities called? What are appropriate unites for T and n (given in the “Words” paragraph)? Answer: The letters represent the variables, T is degrees F and N is trips per minute
b. Assign function letters for the following three relations: the temperature as a function of chirp rate for the field cricket, for the tree cricket, and for the snowy tree cricket. Answer: F,T,S
c. Write the three relations mentioned in question 1b above in function notation using appropriate variables and function letters. Notice we do not have formulas yet, just function notation. T=f(n) T=t(n) T=s(n)
d. For each of the relations written in function notation above, what is the input quantity and variable, and what is the output quantity and variable? Include units. T=f(N) The input is n or trips per minute and the output is T in degrees Fahrenheit
2. Formula: Now let’s find formulas for our functions. That is, we need to find an equation relating temperature to the number of chirps per minute for each type of cricket.
a. Notice that each ‘rule’ above involves counting the number of chirps in a different predetermined period, i.e. 15 seconds for the field cricket, 7 seconds for the tree cricket, and 14 seconds for the snowy tree cricket, but our input is the number of chirps per minute.
i. For the field cricket, if n is the number of chirps per minute, how many chirps are there in fifteen seconds?
ii. For the tree cricket, if n is the number of chirps per minute, how many chirps are there in seven seconds?
iii. For the snowy tree cricket, if n is the number of chirps per minute, how many chirps are there in fourteen seconds?
b. Translate the ‘rule’ for each cricket into an equation where the input is n, number of chirps per minute, and the output is T, temperature in degrees Fahrenheit.
3. Table: Next, we will look at presenting the relation between cricket chirp rate and temperature in tabular form. Don’t forget to pay attention to units. The temperature is measured in degrees Fahrenheit, and the chirp rate is in chirps per minute.
a. For each cricket, create a chart giving (as outputs) the temperature when given (as input) the following data: 0 chirps per minute, 15 chirps per minute, 30 chirps per minute, 45 chirps per minute. You can insert your table below.
b. Which cricket starts chirping (has 0 chirps per minute) at the highest temperature? Explain how you found this.
4. Graph: Finally, we will represent each of these three functions graphically.
a. For each cricket, use Excel to create a scatter plot (graph) of the data on the charts in problem 3. Copy your results here.
b. By using your graphs, estimate (eyeball) the temperature if the field cricket is chirping at a rate of 20 chirps per minute.
c. By using your graphs, estimate the temperature when the snowy tree cricket chirps at 60 chirps per minute.
5. Looking ahead: Do you recognize any similarities in the formulas for these three functions? What types of functions are these? How do you know?