A ramp profile is drawn with x and the height y both in metres (m)…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A ramp profile is drawn with and the height both in metres (m). For the height is , and for the height is . The builder wants the two parts to meet without a jump at m, so that the limit of the height as exists. Find the value of the constant . Then find the height of the ramp at m.
Need a reminder? The key ideaShow
A limit looks at the values of near , never at itself. The limit exists when the left-hand and right-hand limits are equal. For a polynomial, the limit at is its value at .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Given. Left part , right part (both in m).
Plan. Make the left-hand limit equal to the right-hand limit.
Step 1. Right-hand limit. As , .
Step 2. Left-hand limit. As , metres.
Step 3. Make them equal. , so .
Step 4. The height at m. The rule for gives , and it matches the limit.
Check. With m the left part is , which gives at m.
Answer: and the ramp is high at m.
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