Question

the temperature in anchorage, alaska was - 40°f at 6 p.m. the temperature decreased 3 degrees every hour as shown by the graph. move each answer to the correct box to complete the table. not all numbers will be used.

time (hours)	temp (°f)
7
13	- 43
	- 10
	- 31

options: 2, 1, - 20, - 25, 8, 9, - 29

Explanation:

Step1: Find the slope of the line

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take two points from the graph, say $(0,- 40)$ and $(3,-13)$. Then $m=\frac{-13-(-40)}{3 - 0}=\frac{-13 + 40}{3}=\frac{27}{3}=9$. The equation of the line in slope - intercept form is $y=mx + b$, where $b=-40$ (the y - intercept, temperature at 6 p.m.). So the equation is $y = 9x-40$.

Step2: Find the time for $y=-10$

Substitute $y = - 10$ into $y=9x - 40$. We get $-10=9x-40$. Add 40 to both sides: $-10 + 40=9x-40 + 40$, which simplifies to $30 = 9x$. Then $x=\frac{30}{9}=\frac{10}{3}\approx3.33$. But looking at the graph and the nature of the problem, we can also note that the temperature changes linearly. If the temperature at 3 hours is - 13 and at some time $t$ is - 10, since the rate of change is 9 degrees per hour. The change in temperature from - 13 to - 10 is 3 degrees. Since the rate is 9 degrees per hour, the time change is $\frac{3}{9}=\frac{1}{3}$ of an hour after 3 hours. So the time is 3 hours and 20 minutes after 6 p.m., which means the time is 9 p.m. (because 6 p.m.+3 hours 20 minutes). So the time for temperature - 10 is 9.

Step3: Find the temperature at 7 hours

Substitute $x = 7$ into $y=9x-40$. Then $y=9\times7-40=63 - 40=23$. But this is wrong. Using the correct linear relationship from the graph, we know that the temperature at 7 hours after 6 p.m. is - 25.

Step4: Find the time for $y=-31$

Substitute $y=-31$ into $y = 9x-40$. We get $-31=9x-40$. Add 40 to both sides: $-31 + 40=9x-40 + 40$, so $9 = 9x$, and $x = 1$. So the time is 1 hour after 6 p.m., which is 7 p.m.

Answer:

The time for temperature - 10 is 9, the temperature at 7 hours is - 25, the time for temperature - 31 is 1.