To represent the relationship between the time (in minutes) and the temperature (in degrees Fahrenheit) in an equation, we can use the linear equation of the form:
\[ y = mx + b \]
where:
- \( y \) is the temperature,
- \( x \) is the time in minutes,
- \( m \) is the rate of change (temperature increase per minute),
- \( b \) is the initial temperature of the oven (when \( x = 0 \)).
From the points given, we can determine the rate of change (\( m \)). We see that:
- At \( (2, 100) \): After 2 minutes, the temperature is 100°F.
- At \( (6, 220) \): After 6 minutes, the temperature is 220°F.
To find the rate of change, we calculate the slope \( m \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{220 - 100}{6 - 2} = \frac{120}{4} = 30 \]
Thus, the oven heats up at a rate of 30°F per minute.
Since we do not have the initial temperature (\( b \)), we can use one of the points to find it. Let's use the point \( (2, 100) \):
\[ 100 = 30(2) + b \] \[ 100 = 60 + b \] \[ b = 100 - 60 = 40 \]
So, the equation representing the relationship is:
\[ y = 30x + 40 \]
Now, regarding the points:
-
The point \( (2, 100) \) means that after 2 minutes, the oven’s temperature was 100 degrees Fahrenheit.
-
The point \( (6, 220) \) means that after 6 minutes, the oven’s temperature was 220 degrees Fahrenheit.
Using the equation, we can make a prediction. For example, to predict the temperature after 10 minutes:
\[ y = 30(10) + 40 = 300 + 40 = 340 \]
Thus, after 10 minutes, the predicted temperature of the oven would be 340 degrees Fahrenheit.
1 answer