To solve this problem, we need to set up a linear equation that represents Marco's distance from the Grand Canyon as a function of time.
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Identify the Rate of Change: Marco's distance from the Grand Canyon decreases by 150 miles every 3 hours. Therefore, his rate of change (slope) can be calculated as follows:
\[ \text{Rate of Change} = \frac{\text{Change in Distance}}{\text{Change in Time}} = \frac{-150 \text{ miles}}{3 \text{ hours}} = -50 \text{ miles per hour} \]
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Initial Value: After 4 hours, Marco's distance from the Grand Canyon is 200 miles. We need to find the initial distance (when \(x = 0\)). If we denote Marco's distance from the Grand Canyon as \(y\), we have:
- At \(x = 4\): \(y = 200\)
Using the slope-intercept form of the linear equation, we can express it as:
\[ y = mx + b \]
where \(m\) is the rate of change (slope) and \(b\) is the initial value.
Given:
- \(m = -50\)
We can use the information known at \(x = 4\) to determine \(b\):
\[ 200 = -50(4) + b \] \[ 200 = -200 + b \] \[ b = 200 + 200 = 400 \]
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Write the Equation: Now we have both the rate of change and the initial value. The equation that represents Marco's distance from the Grand Canyon as a function of the number of hours he drives is:
\[ y = -50x + 400 \]
Where:
- \(y\) is the distance from the Grand Canyon in miles
- \(x\) is the number of hours driven
- The rate of change is \(-50\) miles per hour (indicating a decrease),
- The initial value (distance when \(x = 0\)) is \(400\) miles.