Question

the data represents the number of minutes people spent driving one week. how many peaks does this histogram have?

Explanation:

Step1: Understand Histogram Peaks

A peak in a histogram is a bar that is higher than its adjacent bars. We examine each bar's height relative to neighbors.

Step2: Analyze Each Bar Group

- First group (40 - 90 minutes): The bar here is tall (height 8), and adjacent bars (next to 90) are shorter (height 0). So this is a peak.
- Second group (140 - 190 minutes): The bar here has height 2, adjacent bars (before 140: height 0; after 190: height 0) are shorter? Wait, no—wait, the bar at 40 - 90 is tall, then 90 - 140 is 0, then 140 - 190 is 2, 190 - 240 is 0, 240 - 290 is 2. Wait, no, re - examine: The first bar (40 - 90) is height 8 (tallest so far). Then 90 - 140: height 0. Then 140 - 190: height 2. Then 190 - 240: height 0. Then 240 - 290: height 2. Wait, but the first bar (40 - 90) is a peak (higher than left (none, since it's the first) and right (90 - 140 is 0)). Then 140 - 190: is it a peak? Its left is 0, right is 0, so it's a peak? Wait, no, a peak is a local maximum. So the first bar (40 - 90) is a local max (higher than right). Then 140 - 190: its left is 0, right is 0, so it's a local max? Wait, no, the height of 140 - 190 is 2, and 240 - 290 is also 2. Wait, maybe I misread. Wait the histogram: x - axis is minutes (40 - 90, 90 - 140, 140 - 190, 190 - 240, 240 - 290). The y - axis is count. The first bar (40 - 90) has height 8 (from 0 to 8). Then 90 - 140: height 0. Then 140 - 190: height 2 (from 0 to 2). Then 190 - 240: height 0. Then 240 - 290: height 2 (from 0 to 2). Wait, but the first bar (40 - 90) is a peak (since it's higher than its right neighbor (90 - 140, height 0)). Then 140 - 190: is it a peak? Its left is 0, right is 0, so it's a local maximum. Similarly, 240 - 290: left is 0, right is none (end), so is it a peak? Wait no, the definition of a peak in a histogram is a bar that is higher than both its left and right neighbors (or one neighbor if it's at the end, but usually local maximum). Wait, the first bar (40 - 90): left neighbor: none (start), right neighbor: 90 - 140 (height 0). So it's a peak. The 140 - 190 bar: left neighbor 90 - 140 (height 0), right neighbor 190 - 240 (height 0). So it's a peak. The 240 - 290 bar: left neighbor 190 - 240 (height 0), right neighbor: none (end). So is it a peak? Wait, but the height of 140 - 190 and 240 - 290 is 2, same. But the first bar is height 8. Wait, maybe I made a mistake. Wait the original histogram: looking at the graph, the first bar (40 - 90) is tall (height 8), then 90 - 140 is 0, then 140 - 190 is 2, 190 - 240 is 0, 240 - 290 is 2. So the peaks are the bars that are higher than their adjacent bars. So the first bar (40 - 90) is a peak (higher than right). The 140 - 190 bar: higher than left (0) and right (0). The 240 - 290 bar: higher than left (0). Wait, no, maybe the question's histogram has three peaks? Wait no, wait the first bar (40 - 90) is one peak. Then 140 - 190 is another (since left is 0, right is 0). Then 240 - 290 is another? Wait no, the height of 140 - 190 and 240 - 290 is same, but are they peaks? Wait, no, maybe I misread the heights. Wait the y - axis: 0,1,2,3,4,5,6,7,8,9,10. The first bar (40 - 90) goes up to 8. Then 90 - 140: 0. Then 140 - 190: up to 2. Then 190 - 240: 0. Then 240 - 290: up to 2. So the peaks are the bars where the height is a local maximum. So the first bar (40 - 90) is a peak (local max, since right is lower). The 140 - 190 bar: local max (left and right lower). The 240 - 290 bar: local max (left lower, right none). Wait, but that would be three? Wait no, maybe the 140 - 190 and 240 - 290 are not peaks because their height is same and not higher than the first? No, peak is about local, not global. Wait, no, let's check again. The first bar: height 8 (neighbor to the right is 0) → peak. The bar at 140 - 190: height 2 (neighbor to the left is 0, neighbor to the right is 0) → peak. The bar at 240 - 290: height 2 (neighbor to the left is 0) → peak. Wait, but that seems like three. Wait, no, maybe the 140 - 190 and 240 - 290 are not considered peaks because they are the same height and not higher than the first, but the definition of a peak in a histogram is a bar that is higher than its immediate neighbors. So for the 140 - 190 bar: left neighbor (90 - 140) is 0, right neighbor (190 - 240) is 0. So it's a peak. For 240 - 290 bar: left neighbor (190 - 240) is 0, right neighbor: none. So it's a peak. And the first bar (40 - 90): left neighbor: none, right neighbor (90 - 140) is 0. So it's a peak. Wait, but that would be three. Wait, maybe I misread the graph. Wait the original graph: the first bar (40 - 90) is height 8. Then 90 - 140: 0. Then 140 - 190: height 2. Then 190 - 240: 0. Then 240 - 290: height 2. So the peaks are the bars with height higher than adjacent. So first bar: peak. 140 - 190: peak. 240 - 290: peak. Wait, but that's three? Wait no, maybe the 140 - 190 and 240 - 290 are not peaks because they are symmetric and the same height, but the definition is local maximum. So each local maximum is a peak. So the answer should be 3? Wait, no, wait the first bar is 40 - 90 (height 8), then 90 - 140 (0), 140 - 190 (2), 190 - 240 (0), 240 - 290 (2). So the local maxima are at 40 - 90 (since it's higher than right), 140 - 190 (higher than left and right), and 240 - 290 (higher than left). So three peaks? Wait, but maybe I made a mistake. Wait, let's check the definition again. A peak in a histogram is a bar that is taller than the bars immediately next to it (on the left and right, or one side if at the end). So:

- 40 - 90: taller than right (90 - 140 is 0) → peak.

- 140 - 190: taller than left (90 - 140 is 0) and right (190 - 240 is 0) → peak.

- 240 - 290: taller than left (190 - 240 is 0) → peak.

So there are 3 peaks? Wait, but the graph shows three groups of bars with height > 0: 40 - 90 (height 8), 140 - 190 (height 2), 240 - 290 (height 2). Each of these groups has a bar that is a local maximum (higher than adjacent bars with height 0). So the number of peaks is 3. Wait, but maybe I misread the heights. Wait the y - axis: the first bar goes up to 8 (from 0 to 8), the second (140 - 190) goes up to 2 (from 0 to 2), the third (240 - 290) goes up to 2 (from 0 to 2). So each of these is a peak because they are higher than their adjacent bars (which are height 0). So the number of peaks is 3.

Answer:

3