Entomologists are biological scientists who study insects. Entomologists studying tree crickets have found that they chirp at different rates depending on the temperature.

The number of chirps per minute, C, that the tree cricket makes is linearly dependent on the temperature, T, in Fahrenheit. The crickets do not chirp at all below 39 degrees and at 73 degrees they chirp about 153 times per minute.

A)  Express the number of chirps, C, as a function of the temperature, T. Be sure to use function notation.  
  

B)  How many chirps per minute will crickets make at 93 degrees?  
 
Select an answer


Let's break this down step by step:

### Part A: Express \( C \) as a function of \( T \)

We know the following:
1. The relationship between chirps \( C \) and temperature \( T \) is **linear**.
2. At \( T = 39^\circ F \), the crickets do not chirp at all, so \( C = 0 \).
3. At \( T = 73^\circ F \), the crickets chirp 153 times per minute, so \( C = 153 \).

We can write the equation of a line in slope-intercept form: 
\[
C = mT + b
\]
where:
- \( C \) is the number of chirps per minute,
- \( T \) is the temperature,
- \( m \) is the slope of the line, and
- \( b \) is the y-intercept (which represents the chirps when \( T = 0 \)).

To find the slope \( m \), we use the two points:
\[
(T_1, C_1) = (39, 0) \quad \text{and} \quad (T_2, C_2) = (73, 153)
\]
The formula for the slope between two points is:
\[
m = \frac{C_2 - C_1}{T_2 - T_1} = \frac{153 - 0}{73 - 39} = \frac{153}{34} = 4.5
\]
So, the slope \( m \) is 4.5.

Now, to find \( b \), we use one of the given points. Let’s use \( (39, 0) \):
\[
C = mT + b \quad \Rightarrow \quad 0 = 4.5(39) + b
\]
\[
0 = 175.5 + b \quad \Rightarrow \quad b = -175.5
\]

Thus, the equation of the line is:
\[
C(T) = 4.5T - 175.5
\]

### Part B: How many chirps per minute will crickets make at \( 93^\circ F \)?

Now, use the function \( C(T) = 4.5T - 175.5 \) and substitute \( T = 93 \):

\[
C(93) = 4.5(93) - 175.5 = 418.5 - 175.5 = 243
\]

So, at \( 93^\circ F \), the crickets will chirp **243 times per minute**.

### Summary:
- **A)** \( C(T) = 4.5T - 175.5 \)
- **B)** At \( 93^\circ F \), the crickets will chirp **243 times per minute**.

Would you like more details or have any further questions?

Here are 5 related questions:
1. What is the significance of the slope in the linear equation in this context?
2. How would the equation change if crickets chirped 50 times at 50 degrees Fahrenheit?
3. At what temperature would the crickets chirp 100 times per minute?
4. How would the equation look if the relationship was quadratic instead of linear?
5. What assumptions are made when using a linear model for this problem?

**Tip:** In linear relationships, the slope indicates the rate of change. In this case, it shows how much the chirping rate increases for each 1-degree increase in temperature.

Entomologists are biological scientists who study insects. Entomologists studying tree crickets have found that they chirp at different rates depending on the temperature. The number of chirps per minute, C, that the tree cricket makes is linearly dependent on the temperature, T, in Fahrenheit. The crickets do not chirp at all below 39 degrees and at 73 degrees they chirp about 153 times per minute.
A) Express the number of chirps, C, as a function of the temperature, T. Be sure to use function notation.
B) How many chirps per minute will crickets make at 93 degrees?

Explore how the chirping rate of crickets changes with temperature using a linear equation. This problem demonstrates how to derive a linear function based on given data points and solve for chirping rates at different temperatures. Ideal for students in Grades 8-10.

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