the brookdale cricket chirps at different rates depending on the temperature. Fo
ID: 3004820 • Letter: T
Question
the brookdale cricket chirps at different rates depending on the temperature. For this cricket, you can estimate the temperature in degrees Fahrenheit by counting the number of times it chirps in 10 seconds and then adding 45.
The brookdale cricket is not the only animal whose activity level increases as the temperature increases. The Green katydid is small, pale green and usually found on trees. for the katydid, you can estimate the temperature by counting the number of chirps in 30 seconds and adding 25.
1a. Notice that the rule for each animal involves counting the number of chirps in different time intervals. For the brookdale cricket you can count the number of chiprs in 10 seconds and for the katydid you count the number of chirps in 30 seconds. If c is the number of chirps per minute, what is the expression for the number of chirps for i) The brookdale cricket in 10 seconds? ii) The katydid in 30 seconds?
1b. Translate the "rule for each animal into a function whose independent variable is c and whose dependent variable is T, temperature in degrees Fahrenheit. Identify which formaula goes with which animal.
1c. If each animal chirps 100 times in a minute, what are the estimates of the temperature for each animal? Show all substitutions into the formulats and identify the animal used to give the estimate.
1e. for each function specify a domain and corresponding range. Assume that most animals no longer chirp when the temperature is above 100 defrees.
1h. if the temperature is 75 degrees Fahrenheit, how many chirps will the brookdale cricket make per minute? write the equation and show how you get this answer in a complete sentence.
3. A highway engineer wants to estimate the maximum number of cars that safely travel a particular highway at a given speed. she assumes that each car is 17 feet long, travels at speed s (miles per hour), and follows the car in front of it at "safe following distance" for that speed. she finds that number N of cars that can pass a given point per minute is model by the function N(s) = 88(s)/17+17(s/20)square.
3a. Evaluate N(45) and N(80) and explain their meaning in the context of the problem. Use complete sentences.
3e. At what speed can the greatest number of cars travel the highway safety?
Explanation / Answer
1 Cricket: Chirps/10 sec = Chirps/Min/6 = c/6
Katydid : = c/2
1b Tc = c/6 + 45, Tk = c/2 + 25
1c) Tc = 100/6 + 45 = 61.67, Tk = 100/2 + 25 = 75
1e) Brookdale Domain [0,330] range is [45,100], Katydid Domain [0,150], Range [25,100]
1h) 75 = c/6 + 45 => c =180 chirps/min
3a) N(s) = 88s/17 - 17s2/400
N(45) = 88*45/17 - 17*452/400 = 147 cars
N(80) = 142 cars.
This indicates that more the speed, less cars can travel safely.
3e) Take the first derivative and set it to zero for finding optimal speed to get greatest number of cars
88/17 - 17s/200 = 0 => s = 88*200/289 = 60.9 mph
Note: I assumed a minus sign for the 2nd term in the N(s) equation as it makes logical sense.
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