Solving Linear Function: Temperature & Chirp Rate

In summary: It gives 77.2 as the temperature at 150 chirps per minute, which is closer to the textbook answer of 76°F than the calculated -26°F. So the textbook is most likely correct.
  • #1
physics604
92
2
I know how to do this question, but I'm trying to figure out if the textbook answer is wrong. This is from James Stewart's Calculus Early Transcendentals, 7th edition.

1. Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70°F and 173 chirps per minute at 80°F.

(a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N.
(b) What is the slope of the graph? What does it represent?
(c) If the crickets are chirping at 150 chirps per minute, estimate the temperature.



Homework Equations



y=mx+b

The Attempt at a Solution



a) The question wants T as a function of time (don't ask me why it's weird like that), so I know my equation has to be T=mN+b. After punching (113,70) and (173,80) into my calculator, I get y=1/6N-307/6. This is what the textbook had. Great. Please keep going.

b) Slope is 1/6, easy. It represents change in for every chirp per minute change. This was right. Please read on one more.

c) It wants the temperature when N=150. The problem is, using the equation, the temperature would be about -26°F. The textbook said this was wrong. Their answer was 76°F. The only way you can get 76 as an answer is if your initial equation was y=6N-307 (I know because I used this equation first).

Am I right? Is the textbook wrong?
 
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  • #2
physics604 said:
I know how to do this question, but I'm trying to figure out if the textbook answer is wrong. This is from James Stewart's Calculus Early Transcendentals, 7th edition.

1. Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70°F and 173 chirps per minute at 80°F.

(a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N.
(b) What is the slope of the graph? What does it represent?
(c) If the crickets are chirping at 150 chirps per minute, estimate the temperature.



Homework Equations



y=mx+b

The Attempt at a Solution



a) The question wants T as a function of time (don't ask me why it's weird like that), so I know my equation has to be T=mN+b. After punching (113,70) and (173,80) into my calculator, I get y=1/6N-307/6. This is what the textbook had. Great. Please keep going.
You mean chirp rate, N, not time.

b) Slope is 1/6, easy. It represents change in for every chirp per minute change. This was right. Please read on one more.

c) It wants the temperature when N=150. The problem is, using the equation, the temperature would be about -26°F. The textbook said this was wrong. Their answer was 76°F. The only way you can get 76 as an answer is if your initial equation was y=6N-307 (I know because I used this equation first).
Show your work. You're either setting the problem up wrong or solving for T incorrectly.

Am I right? Is the textbook wrong?
The text appears to be right.
 
  • #3
physics604 said:
a) so I know my equation has to be T=mN+b. After punching (113,70) and (173,80) into my calculator, I get y=1/6N-307/6. This is what the textbook had. Great. Please keep going.

If y=T, the equation is wrong. It has to be y=1/6 N+307/6, that is T=1/6 N+307/6, or

N=6T-307.

ehild
 
  • #4
You can always check c) from the OP by doing linear interpolation on the two data points given.
 

FAQ: Solving Linear Function: Temperature & Chirp Rate

1. What is a linear function?

A linear function is a mathematical function that can be represented by a straight line. It is of the form y = mx + b, where x is the independent variable, y is the dependent variable, m is the slope of the line, and b is the y-intercept.

2. How do temperature and chirp rate relate to a linear function?

In the context of crickets, temperature and chirp rate follow a linear relationship. As the temperature increases, the chirp rate also increases in a predictable pattern. This can be represented by a linear function where the temperature is the independent variable and the chirp rate is the dependent variable.

3. How do you solve a linear function for temperature and chirp rate?

To solve a linear function for temperature and chirp rate, you will need to have two data points that represent the temperature and corresponding chirp rate. Then, you can use the slope-intercept form of the linear function (y = mx + b) to find the slope (m) and y-intercept (b). Once you have these values, you can plug them into the linear function equation to find the temperature for a given chirp rate, or vice versa.

4. Can a linear function accurately predict temperature and chirp rate?

Yes, a linear function can accurately predict temperature and chirp rate as long as the data points used to create the function are accurate and follow a linear relationship. However, it is important to note that there may be other factors that can affect the relationship between temperature and chirp rate, such as humidity or the age of the cricket.

5. How can solving linear functions for temperature and chirp rate be useful?

Solving linear functions for temperature and chirp rate can be useful in predicting the temperature based on the chirp rate of crickets. This can be helpful for researchers studying cricket behavior and for predicting weather patterns based on the chirping of crickets. It can also be a useful skill for students learning about linear functions and their applications in real-world scenarios.

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