Question

1. when bethany goes to bed at night, she turns the heat off in her house. when she went to bed last night, the temperature in her house was 71° fahrenheit. when she woke up 8 hours later, the temperature in her house was 55°f. assume that the temperature in her house drops at a constant rate.

a. graph the relationship between the temperature in bethanys house and the hours since she went to bed.
b. what is the y -intercept of the graphed line? what does the y -intercept mean in the context of this problem?
c. what is the slope of the graphed line? what does the slope mean in the context of this problem?

Explanation:

Step1: Identify two - point form

We know that at $t = 0$ (when she went to bed), the temperature $T=71^{\circ}F$ and at $t = 8$ hours, $T = 55^{\circ}F$. The two - points are $(0,71)$ and $(8,55)$.

Step2: Find the slope formula

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $x_1 = 0,y_1=71,x_2 = 8,y_2 = 55$. So $m=\frac{55 - 71}{8-0}=\frac{- 16}{8}=-2$.

Step3: Find the equation of the line

The equation of a line in slope - intercept form is $y=mx + b$, where $m$ is the slope and $b$ is the $y$ - intercept. Since the line passes through $(0,71)$, when $x = 0,y=71$, so $b = 71$. The equation of the line is $y=-2x + 71$.

Step4: Graph the line

To graph the line $y=-2x + 71$:

• The $y$ - intercept is 71, so plot the point $(0,71)$.
• Using the slope $m=-2=\frac{\Delta y}{\Delta x}$, from the point $(0,71)$, move 1 unit to the right (increase $x$ by 1) and 2 units down (decrease $y$ by 2) to get another point $(1,69)$. Connect the points to draw the line.

Step5: Answer part b

The $y$ - intercept of the graphed line is 71. In the context of the problem, it represents the temperature of the house (in degrees Fahrenheit) when Bethany went to bed (at time $x = 0$ hours).

Step6: Answer part c

The slope of the graphed line is - 2. In the context of the problem, it means that the temperature of the house drops 2 degrees Fahrenheit per hour.

Answer:

a. Graph the line $y=-2x + 71$ using the $y$ - intercept $(0,71)$ and slope $m=-2$.
b. The $y$ - intercept is 71. It represents the initial temperature of the house when Bethany went to bed.
c. The slope is - 2. It represents the rate at which the temperature of the house drops per hour (2 degrees Fahrenheit per hour).