Answer :
Sure! Let's determine whether the function representing the bicycle rental cost as a function of time in hours represents a direct variation. Here's a step-by-step analysis:
1. Definition of Direct Variation: A function represents a direct variation if it can be described by the equation \( y = kx \), where \( k \) is a constant. This means that the function passes through the origin (0,0) and has a constant rate of change (slope).
2. Given Values:
- Time (hours): \(0\), \( 2\), \( 4\), \( 6\), \( 8\)
- Cost (\$): \(0\), \( 10\), \( 20\), \( 30\), \( 40\)
3. Checking if it Passes Through the Origin:
- The function must have a cost of \$0 at time 0 hours.
- From the table, we see that at time 0 hours, the cost is \$0.
4. Calculating the Rate of Change:
- The rate of change is the change in cost divided by the change in time (the slope).
- Calculate the rate of change between consecutive points:
- From 0 to 2 hours: \( \frac{10 - 0}{2 - 0} = \frac{10}{2} = 5 \) dollars per hour
- From 2 to 4 hours: \( \frac{20 - 10}{4 - 2} = \frac{10}{2} = 5 \) dollars per hour
- From 4 to 6 hours: \( \frac{30 - 20}{6 - 4} = \frac{10}{2} = 5 \) dollars per hour
- From 6 to 8 hours: \( \frac{40 - 30}{8 - 6} = \frac{10}{2} = 5 \) dollars per hour
- The rate of change is consistent and equal to 5 dollars per hour.
5. Conclusion:
- Since the function passes through the origin and has a constant rate of change of $5 per hour, it represents a direct variation.
Thus, the correct explanation is:
1. This function represents a direct variation because it passes through the origin and has a constant rate of change of $5 per hour.
1. Definition of Direct Variation: A function represents a direct variation if it can be described by the equation \( y = kx \), where \( k \) is a constant. This means that the function passes through the origin (0,0) and has a constant rate of change (slope).
2. Given Values:
- Time (hours): \(0\), \( 2\), \( 4\), \( 6\), \( 8\)
- Cost (\$): \(0\), \( 10\), \( 20\), \( 30\), \( 40\)
3. Checking if it Passes Through the Origin:
- The function must have a cost of \$0 at time 0 hours.
- From the table, we see that at time 0 hours, the cost is \$0.
4. Calculating the Rate of Change:
- The rate of change is the change in cost divided by the change in time (the slope).
- Calculate the rate of change between consecutive points:
- From 0 to 2 hours: \( \frac{10 - 0}{2 - 0} = \frac{10}{2} = 5 \) dollars per hour
- From 2 to 4 hours: \( \frac{20 - 10}{4 - 2} = \frac{10}{2} = 5 \) dollars per hour
- From 4 to 6 hours: \( \frac{30 - 20}{6 - 4} = \frac{10}{2} = 5 \) dollars per hour
- From 6 to 8 hours: \( \frac{40 - 30}{8 - 6} = \frac{10}{2} = 5 \) dollars per hour
- The rate of change is consistent and equal to 5 dollars per hour.
5. Conclusion:
- Since the function passes through the origin and has a constant rate of change of $5 per hour, it represents a direct variation.
Thus, the correct explanation is:
1. This function represents a direct variation because it passes through the origin and has a constant rate of change of $5 per hour.
