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<h3>1. How many inches per minute does London's elevation change between 4 minutes and 8 minutes. </h3>
The question actually asks for the slope of the line that stands for the points why? because the questions tells us that London's elevation changes between 4 minutes and 8 minutes here. Hence, to find the slope of this line we have to use the following formula:
<em>So London's elevation changes 0.75 inches per minute</em>
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<h3>2. During which time period does London's elevation change the fastest?</h3>The greater the absolute value of the slope of the line the faster London's elevation changes. Since this is a Piecewise function, we must analyze each period.
- FIRST:
→ Between 0 minutes and 4 minutes the function is constant, so there is no any change here.
→ Between 10 minutes and 14 minutes the function is constant, so there is no any change here.
→ Between 18 minutes and 22 minutes the function is constant, so there is no any change here.
So the solution is not in these parts of the function.
- SECOND:
→ Between 4 minutes and 10 minutes the function has a positive slope, so there is change here.
In the previous item we calculated the slope between 4 and 8 minutes that is the same slope between 4 and 8 minutes and equals 0.75.
→ Between 14 minutes and 18 minutes the function has a positive slope, so there is change here.
Let's take two points here, say,
As you can see, the absolute value here is 1 that is greater than 0.75.
<em>In conclusion, London's elevation changes the fastest between 14 and 18 minutes</em>
You can use the sum of angles identities, then rearrange to put the result in the form of tangents.