Question

Time (hours) 0 4 7 9 12 Distance (miles) 0 248 449 565 757 The Diaz family went on a road trip for vacation. The table includes a cumulative chart of the time they traveled compared to the distance they drove. Over which interval was their average speed the slowest and fastest: [0, 4], [4, 7], [7, 9], or [9, 12]? Show work for each interval's average speed. Slowest interval: [, ] Fastest interval: [, ]

Answer 1

To find the average speed over each interval, we use the formula for average speed:

\[ \text{Average Speed} = \frac{\text{Change in Distance}}{\text{Change in Time}} \]

We will calculate the average speed for each given interval:

1. Interval [0, 4]

   • Change in Distance: \(248 - 0 = 248\) miles
   • Change in Time: \(4 - 0 = 4\) hours
   • Average Speed: \[ \frac{248 \text{ miles}}{4 \text{ hours}} = 62 \text{ miles per hour} \]

2. Interval [4, 7]

   • Change in Distance: \(449 - 248 = 201\) miles
   • Change in Time: \(7 - 4 = 3\) hours
   • Average Speed: \[ \frac{201 \text{ miles}}{3 \text{ hours}} = 67 \text{ miles per hour} \]

3. Interval [7, 9]

   • Change in Distance: \(565 - 449 = 116\) miles
   • Change in Time: \(9 - 7 = 2\) hours
   • Average Speed: \[ \frac{116 \text{ miles}}{2 \text{ hours}} = 58 \text{ miles per hour} \]

4. Interval [9, 12]

   • Change in Distance: \(757 - 565 = 192\) miles
   • Change in Time: \(12 - 9 = 3\) hours
   • Average Speed: \[ \frac{192 \text{ miles}}{3 \text{ hours}} = 64 \text{ miles per hour} \]

Now, let's summarize the average speeds:

• Average speed for [0, 4]: 62 mph
• Average speed for [4, 7]: 67 mph
• Average speed for [7, 9]: 58 mph
• Average speed for [9, 12]: 64 mph

From the calculations:

• The slowest average speed is 58 mph over the interval [7, 9].
• The fastest average speed is 67 mph over the interval [4, 7].

Thus, we conclude:

Slowest interval: [7, 9]
Fastest interval: [4, 7]