Question

Sara is baking cookies, so she turns the oven on. As soon as Sara turns the oven on, the oven heats up at a constant rate. Write an equation to represent this situation. Then use the equation to make a prediction.
This graph represents the linear relationship between the time in minutes and the temperature of the oven in degrees Fahrenheit. For each minute, the temperature increases at a constant rate. What do the points (4, 120) and (14, 340) represent?
x
y
(4, 120)(14, 340)
0
2
4
6
8
10
12
14
16
18
20
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
380
400
Time (minutes)
Temperature (°F)
The point (4, 120) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.

The point (14, 340) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.

Answer 1

To model the situation, we can write a linear equation that relates the time (x) in minutes to the temperature (y) in degrees Fahrenheit. Since the temperature increases at a constant rate, we can express the relationship as:

\[ y = mx + b \]

Where:
- \( y \) is the temperature in degrees Fahrenheit.
- \( x \) is the time in minutes.
- \( m \) is the rate of temperature increase per minute.
- \( b \) is the initial temperature of the oven when \( x = 0 \).

From the two points given:
1. (4, 120): at 4 minutes, the temperature is 120°F.
2. (14, 340): at 14 minutes, the temperature is 340°F.

### Step 1: Find the rate of temperature increase (m)

We can find \( m \) using the formula for slope:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points (4, 120) and (14, 340):
- \( y_1 = 120 \), \( x_1 = 4 \)
- \( y_2 = 340 \), \( x_2 = 14 \)

\[
m = \frac{340 - 120}{14 - 4} = \frac{220}{10} = 22
\]

So, the oven temperature increases at a rate of 22°F per minute.

### Step 2: Find the initial temperature (b)

To find \( b \), we can use one of the points in the equation. We'll use the point (4, 120):

\[
y = mx + b
\]
\[
120 = 22(4) + b
\]
\[
120 = 88 + b
\]
\[
b = 120 - 88 = 32
\]

### The equation for the temperature of the oven:

Thus, our linear equation is:

\[
y = 22x + 32
\]

### Interpret the given points

Now we interpret the points:
- The point (4, 120) means that after **4 minutes**, the oven’s temperature was **120 degrees Fahrenheit**.

- The point (14, 340) means that after **14 minutes**, the oven’s temperature was **340 degrees Fahrenheit**.

Feel free to ask if you have any further questions!