Let's summarize the key points and calculations made in this problem.
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Understanding the points:
- 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.
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Finding the slope (m): To find the slope \( m \), we can use the formula: \[ m = \frac{\text{change in } y}{\text{change in } x} = \frac{350 - 150}{12 - 4} = \frac{200}{8} = 25 \] Thus, the slope \( m = 25 \) degrees per minute.
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Setting up the equation: Given the slope \( m \) and knowing that the equation is in the form \( y = mx + b \): \[ y = 25x + b \]
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Finding the y-intercept (b): Using the point (4, 150): \[ 150 = 25(4) + b \ 150 = 100 + b \ b = 150 - 100 = 50 \] Therefore, the y-intercept \( b = 50 \).
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Final equation: The equation representing the relationship between time and the oven temperature is: \[ y = 25x + 50 \]
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Predicting temperature after 14 minutes: Now, substitute \( x = 14 \) into the equation: \[ y = 25(14) + 50 \ y = 350 + 50 = 400 \] So, the predicted temperature of the oven after 14 minutes is 400 degrees Fahrenheit.
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Comparing prediction to actual data: According to the graph, if we check the actual temperature recorded after 14 minutes, we find that it is 400 degrees Fahrenheit as well.
Thus, our prediction for 14 minutes is the same as the actual data on the graph.