Question

1. mai hikes up a trail for 40 minutes. the graph shows the elevation in feet that she reaches throughout her hike. name the time period where mai gains elevation at the fastest rate.

graph with elevation (feet) on y - axis from 0 to 240 and time (minutes) on x - axis from 0 to 40, showing mais elevation over time

Explanation:

Step1: Understand rate of elevation gain

The rate of elevation gain is the slope of the elevation - time graph. A steeper slope means a faster rate of elevation gain. The slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(y\) is elevation (in feet) and \(x\) is time (in minutes).

Step2: Analyze different intervals

• Interval 0 - 22 minutes: Let's take two points, say \((0,0)\) and \((22,110)\) (approximate from the graph). The slope \(m_1=\frac{110 - 0}{22 - 0}=\frac{110}{22} = 5\) feet per minute.
• Interval 22 - 26 minutes: Take points \((22,110)\) and \((26,170)\) (approximate). The slope \(m_2=\frac{170 - 110}{26 - 22}=\frac{60}{4}=15\) feet per minute.
• Interval 26 - 40 minutes: Take points \((26,170)\) and \((40,240)\) (approximate). The slope \(m_3=\frac{240 - 170}{40 - 26}=\frac{70}{14} = 5\) feet per minute.

By comparing the slopes of different intervals, we can see that the slope (rate of elevation gain) is highest in the interval 22 - 26 minutes.

Answer:

The time period where Mai gains elevation at the fastest rate is from 22 minutes to 26 minutes.