Question
Derek is baking cookies, so he turns the oven on. As soon as Derek 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, 150) and (12, 350) represent?
x
y
(4, 150)(12, 350)
0
2
4
6
8
10
12
14
16
18
20
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
Time (minutes)
Temperature (°F)
The point (4, 150) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.
The point (12, 350) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.
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, 150) and (12, 350) represent?
x
y
(4, 150)(12, 350)
0
2
4
6
8
10
12
14
16
18
20
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
Time (minutes)
Temperature (°F)
The point (4, 150) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.
The point (12, 350) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.
Answers
GPT-4o mini
To model the situation with an equation, we can let \( y \) represent the temperature of the oven in degrees Fahrenheit and \( x \) represent the time in minutes. Since the oven heats up at a constant rate, we can express the relationship as a linear equation in the form:
\[ y = mx + b \]
where \( m \) is the rate of change (slope), and \( b \) is the initial temperature of the oven when \( x = 0 \).
Given the points (4, 150) and (12, 350):
1. **Calculate the slope \( m \)**:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{350 - 150}{12 - 4} = \frac{200}{8} = 25
\]
This means the temperature increases by 25 degrees Fahrenheit per minute.
2. **Find the y-intercept \( b \)**:
We can use one of the points to find \( b \). Using point (4, 150):
\[
150 = 25(4) + b \\
150 = 100 + b \\
b = 150 - 100 = 50
\]
Now, we can write the equation for the temperature of the oven:
\[ y = 25x + 50 \]
### Predictions:
- The point **(4, 150)** means that **after 4 minutes, the oven’s temperature was 150 degrees Fahrenheit**.
- The point **(12, 350)** means that **after 12 minutes, the oven’s temperature was 350 degrees Fahrenheit**.
### Using the equation to make a prediction:
To predict the temperature after, say, 10 minutes:
\[
y = 25(10) + 50 \\
y = 250 + 50 = 300
\]
So, after 10 minutes, the oven’s temperature is predicted to be **300 degrees Fahrenheit**.
\[ y = mx + b \]
where \( m \) is the rate of change (slope), and \( b \) is the initial temperature of the oven when \( x = 0 \).
Given the points (4, 150) and (12, 350):
1. **Calculate the slope \( m \)**:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{350 - 150}{12 - 4} = \frac{200}{8} = 25
\]
This means the temperature increases by 25 degrees Fahrenheit per minute.
2. **Find the y-intercept \( b \)**:
We can use one of the points to find \( b \). Using point (4, 150):
\[
150 = 25(4) + b \\
150 = 100 + b \\
b = 150 - 100 = 50
\]
Now, we can write the equation for the temperature of the oven:
\[ y = 25x + 50 \]
### Predictions:
- The point **(4, 150)** means that **after 4 minutes, the oven’s temperature was 150 degrees Fahrenheit**.
- The point **(12, 350)** means that **after 12 minutes, the oven’s temperature was 350 degrees Fahrenheit**.
### Using the equation to make a prediction:
To predict the temperature after, say, 10 minutes:
\[
y = 25(10) + 50 \\
y = 250 + 50 = 300
\]
So, after 10 minutes, the oven’s temperature is predicted to be **300 degrees Fahrenheit**.