Question

The outside temperature can be estimated based on how fast crickets chirp. At 104 chirps per minute, the temperature is 63 degreeree F At 176 chirps per minute, the temperature is 81 degreeree 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 124 chirps per minute? What is the temperature if you hear 68 chirps per minute?

Answer 1

For 124 chirps per minute the temperature is 68 ºF.

For 68 chirps per minute the temperature is 54 ºF.

Explanation:

Linear functions are those whose graph is a straight line. A linear function has the following form

`f(x)=b+mx`

b is the constant term or the y intercept. It is the value of the dependent variable when x = 0.

m is the coefficient of the independent variable. It is also known as the slope and gives the rate of change of the dependent variable.

We know that

• At 104 chirps per minute, the temperature is 63 ºF.

• At 176 chirps per minute, the temperature is 81 ºF.

This information can be converted to Cartesian coordinates (x, y). Where x = the number of chirps per minute and y = the temperature in ºF.

To find a linear function that let us find the outside temperature from how fast crickets chirp we must:

• Find the slope:

`m=(y_2-y_1)/(x_2-x_1)=(81-63)/(176-104)=(1)/(4)`

• Find the equation:

`81=(1)/(4)\cdot 104+b`

Solving for b

`b=81-(1)/(4) (176)=37`

Therefore, the linear function is

`y=(1)/(4) \cdot x+37`

Now, using this linear function we can know the temperature when we know the chirps per minute:

For 124 chirps per minute the temperature is:

`y=(1)/(4) \cdot (124)+37=68`

For 68 chirps per minute the temperature is:

`y=(1)/(4) \cdot (68)+37=54`