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The given problem provides two key points describing the relationship between the number of cricket chirps per minute \( N \) and the temperature \( T \) in degrees Fahrenheit. 

The points are:
1. At \( N = 117 \) chirps/min, \( T = 72^\circ \) Fahrenheit.
2. At \( N = 175 \) chirps/min, \( T = 89^\circ \) Fahrenheit.

The relationship is said to be linear, so we can model it with the equation of a line in the form:

\[
T(N) = mN + b
\]

where:
- \( m \) is the slope,
- \( b \) is the y-intercept.

### (a) Find the linear equation for \( T(N) \):

First, we calculate the slope \( m \) using the two points \((117, 72)\) and \((175, 89)\). The slope is given by:

\[
m = \frac{T_2 - T_1}{N_2 - N_1} = \frac{89 - 72}{175 - 117} = \frac{17}{58}
\]

Now we substitute \( m = \frac{17}{58} \) into the equation, along with one of the points (e.g., \( N_1 = 117 \), \( T_1 = 72 \)) to find the intercept \( b \):

\[
72 = \frac{17}{58} \cdot 117 + b
\]

Solving for \( b \):

\[
72 = \frac{1989}{58} + b
\]
\[
b = 72 - \frac{1989}{58} = 72 - 34.29 = 37.71
\]

Thus, the linear equation is:

\[
T(N) = \frac{17}{58} N + 37.71
\]

### (b) Estimate the temperature when \( N = 160 \):

We substitute \( N = 160 \) into the equation:

\[
T(160) = \frac{17}{58} \cdot 160 + 37.71
\]

\[
T(160) = \frac{2720}{58} + 37.71 = 46.90 + 37.71 = 84.61^\circ \text{F}
\]

Thus, the estimated temperature is approximately **84.61°F**.

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the significance of the slope in this context?
2. How would you modify the equation if the temperature were measured in Celsius instead?
3. If the cricket chirped 200 times per minute, what would the temperature be?
4. What assumptions are made in modeling this relationship as linear?
5. Can you derive a similar model using a different biological phenomenon?

**Tip:** When solving for the slope, always remember to subtract the values in the same order for both the dependent and independent variables.

Biologists have noticed that the chirping of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 117 chirps per minute at 72 degrees Fahrenheit and 175 chirps per minute at 89 degrees Fahrenheit. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) If the crickets are chirping at 160 chirps per minute, estimate the temperature.

This problem involves finding a linear equation to model the temperature in relation to cricket chirps per minute. With 117 chirps at 72°F and 175 chirps at 89°F, you will learn to calculate the slope and find the equation. Additionally, estimate the temperature at 160 chirps per minute.

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