A designer draws the curve y = √(x² + 6x) for x > 0, with lengths in metres (m), and the straight…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A designer draws the curve for , with lengths in metres (m), and the straight line beside it. The curve stays above the line, and the vertical gap between them is . The designer wants to know how wide the gap becomes far from the origin. Find the gap at m, to two decimal places. Then find the value that the gap approaches as tends to infinity. Do not subtract two huge numbers: rationalise instead.
Need a reminder? The key ideaShow
When substitution gives , factorise or rationalise, cancel the common factor, and substitute again. For divide the top and the bottom by the highest power of , and use .
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. in metres, for .
Plan. Rationalise, then divide by the highest power of .
Step 1. Rationalise. .
Step 2. At m. .
Step 3. Divide by . For , .
Step 4. The limit. As , , so .
Check. The gap at m is just below m. The program in the next section shows it creeping up towards m.
Answer: the gap is at m, and it approaches .
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