Question

the outside temperature can be estimated based on how fast crickets chirp.
at 104 chirps per minute, the temperature is 63 °f.
at 176 chirps per minute, the temperature is 81 °f.
using this information, you can make a formula that relates chirp rate to temperature. assume the relationship is linear, that is the points form a straight line when plotted on a graph.
what is the temperature if you hear 120 chirps per minute?
temperature:
what is the temperature if you hear 68 chirps per minute?
temperature:

Explanation:

Step1: Find the slope of the line

Let \( x \) be the number of chirps per minute and \( y \) be the temperature in \(^\circ\text{F}\). We have two points \((x_1, y_1) = (104, 63)\) and \((x_2, y_2) = (176, 81)\). The slope \( m \) is calculated as \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{81 - 63}{176 - 104}=\frac{18}{72}=\frac{1}{4} \).

Step2: Find the equation of the line

Using the point - slope form \( y - y_1=m(x - x_1) \), with \( (x_1,y_1)=(104,63) \) and \( m = \frac{1}{4} \), we get \( y-63=\frac{1}{4}(x - 104) \). Simplifying, \( y-63=\frac{1}{4}x-26 \), so \( y=\frac{1}{4}x + 37 \).

Step3: Calculate temperature for 120 chirps

Substitute \( x = 120 \) into \( y=\frac{1}{4}x + 37 \). \( y=\frac{1}{4}(120)+37=30 + 37 = 67 \).

Step4: Calculate temperature for 68 chirps

Substitute \( x = 68 \) into \( y=\frac{1}{4}x + 37 \). \( y=\frac{1}{4}(68)+37 = 17+37=54 \).

Answer:

For 120 chirps per minute, the temperature is \( 67^\circ\text{F} \).
For 68 chirps per minute, the temperature is \( 54^\circ\text{F} \).