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}
$


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