Single Variable Calculus, Chapter 1, 1.2, Section 1.2, Problem 15

According to a study, the chirping rate of crikets of a certain species is related to temperature as what biologists have noticed. Assuming that the relationship appear to be very linear. A criket produces 113 chirps per minute at 70 degrees $F$ and 173 chirps per minute at $80^\circ F$.

a.) Express the temperature $T$ as a function of the number of chirps per minute N by finding an appropriate linear equation for the given data


$\begin{equation} \begin{aligned} T &= aN + k; \, \text{ where}\\ T &= \text{temperature in} ^\circ F\\ N &= \text{number of chips per minute}\\ a &= \text{corresponding change}\\ k &= \text{constant} \end{aligned} \end{equation}$


Given:

$\begin{equation} \begin{aligned} N &= 133 \text{ when } T = 70\\ N &= 173 \text{ when } T = 80 \end{aligned} \end{equation}$


*Substituting these values to the equation will result to:


$\begin{equation} \begin{aligned} a &= \displaystyle \frac{1}{6}\\ k &= \displaystyle \frac{307}{6} \end{aligned} \end{equation}$



$\boxed{.: T = \displaystyle \frac{N}{6} + \displaystyle \frac{307}{6}}$


b.) Find the slope of the graph then state what does it represent.

The slope is $\displaystyle \frac{1}{6}$, it represents the change in $^\circ F$ fro every chip per minute change.

c.) Estimate the temperature if the crickets are chirping at 150 chirps per minute.


$\begin{equation} \begin{aligned} T = \displaystyle \frac{N}{6} + \displaystyle \frac{307}{6}; \text{ if } N = 150\\ T = \displaystyle \frac{150}{6} + \displaystyle \frac{307}{6} = 76.17 ^\circ F \end{aligned} \end{equation}$


$\boxed{.: T = \approx 76^\circ F}$