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.