Question

Biologists have observed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 120 chirps per minute at ${70}^{\circ }F$ and 168 chirps per minute at ${80}^{\circ }F.$

(a)Find the linear equation that relates the temperature tand the number of chirps per minute n

(b)If the crickets are chirping at 150 chirps per minute, estimate the temperature.

Short answer

(a) The linear equation is $100-48t=-2160.$

(b) The Cricket produces 150 chirps per minute at ${76.25}^{\circ }F.$

Step-by-step solution

Part a. Step 1. Concept of Linear Equation.

A linear equation is an algebraic equation of the form $y=mx+b,$ involving only a constant and a first order linear term, where slope is m and y-intercept is b.

Occasionally, the above equation is called a linear equation in two variables where x and y are the variables.

Part a. Step 2. Two point form of linear equation.

If two points $A(x_1,y_1)$and $B(x_2,y_2)$are given, the equation of $AB$ is$\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right).$

Part a. Step 3. Forming the linear equation.

Let $n_1=120,t_1={70}^{\circ }$

$n_2=168,t_2={80}^{\circ }$

Then the linear equation that relates the temperature t and the number of chirps per minute n

$$\begin{gathered} t-70=\frac{80-70}{168-120}\left(n-120\right) \\ t-70=\frac{10}{48}\left(n-120\right) \\ 48t-3360=10n-1200 \\ 10n-48t=-2160 \end{gathered}$$

Part b. Step 1. Concept of Linear Equation.

A linear equation is an algebraic equation of the form $y=mx+b,$ involving only a constant and a first order linear term, where slope is m and y-intercept is b.

Occasionally, the above equation is called a linear equation in two variables where x and y are the variables.

Part b. Step 2. Concept to determine two points.

If the value of x is given, then we have to evaluate the y by substituting the value in the equation and solve for y.

Part b. Step 3. Estimating the temperature.

The given linear equation is

$10n-48t=-2160$

Substituting$n=150$, we get

$$\begin{gathered} 10\left(150\right)-48t=-2160 \\ -48t=-2160-1500 \\ -48t=-3660 \\ t={76.25}^{\circ } \end{gathered}$$

Hence, the Cricket produces 150 chirps per minute at${76.25}^{\circ }F.$