Chapter 5.7, Problem 3LC

To determine

To find: the average maximum daily temperature in January at latitude of ${50}^{\text{O}}\text{N}$

Answer to Problem 3LC

The average maximum daily temperature in January at latitude of ${50}^{\text{O}}\text{N}$ is $y={20.33}^{\text{O}}\text{F}$

Explanation of Solution

Given info:

The average maximum daily temperature in January for northern latitudes is given below

Latitude $\left({}^{\text{O}}\text{N}\right)$	35	33	30	25	43	40	39
Temperature $\left({}^{\text{O}}\text{F}\right)$	46	52	67	76	32	37	44

Formula used:

Show the slope-point form as follows:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Here, $m$ is the slope, $x,x_2,y_1,y_2$ are coordinate points in the graph.

Show the straight line equation as follows:

  $y=mx+c$

Here, $m$ is the slope, $x$ is the variable, and $c$ is the y-intercept.

Show the slope-intercept form as follows:

  $\left(y-y_1\right)=m\left(x-x_1\right)$

Here, $m$ is the slope, $x,y$ are the variables,and $x_1,y_1$ are the coordinate points

Calculation:

Plot the latitudes and temperature in the graph and draw the trend line which should connect at least 3 points in the trend line to find the slope.

  

Find the slope with two points $\left(40,37\right)$ and $\left(43,32\right)$ using slope-point form as follows:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{32-37}{43-40}\\ m=\frac{-5}{3}\end{array}$

Find the slope-intercept form using $m=\frac{-5}{3}$ and coordinate point $\left(40,37\right)$ as follows:

  $\begin{array}{l}\left(y-y_1\right)=m\left(x-x_1\right)\\ \left(y-37\right)=-\frac{5}{3}\left(x-40\right)\\ 3\left(y-37\right)=-5\left(x-40\right)\\ 3y-111=-5x+200\end{array}$

  $\begin{array}{l}3y=-5x+200+111\\ 3y=-5x+311\\ y=-\frac{5x}{3}+103.66\end{array}$

Hence, the straight line equation is $y=-\frac{5x}{3}+103.66$ .

Find the average maximum daily temperature $\left(y\right)$ with $x={50}^{\text{O}}\text{N}$ as follows:

  $\begin{array}{l}y=-\frac{5x}{3}+103.66\\ y=-\frac{\left(5\times 50\right)}{3}+103.66\\ y=-83.33+103.66\\ y=20.33\end{array}$

Hence, the average maximum daily temperature in January at latitude of ${50}^{\text{O}}\text{N}$ is $y={20.33}^{\text{O}}\text{F}$ .

Conclusion:

Thus, the average maximum daily temperature in January at latitude of ${50}^{\text{O}}\text{N}$ is $y={20.33}^{\text{O}}\text{F}$ .