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At the first stage of a pattern, a point moves 4 units from the origin in the positive direction along the x-axis - Leaving Cert Mathematics - Question 9 - 2016

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At the first stage of a pattern, a point moves 4 units from the origin in the positive direction along the x-axis. For the second stage, it turns left and moves 2 un... show full transcript

Worked Solution & Example Answer:At the first stage of a pattern, a point moves 4 units from the origin in the positive direction along the x-axis - Leaving Cert Mathematics - Question 9 - 2016

Step 1

How many stages has the point completed when the total distance it has travelled, along its path, is 7.9375 units?

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Answer

To find the total distance traveled, we can look at the pattern:

  1. Stage 1: 4 units
  2. Stage 2: 2 units
  3. Stage 3: 1 unit
  4. Stage 4: 0.5 units
  5. Stage 5: 0.25 units
  6. Stage 6: 0.125 units
  7. Stage 7: 0.0625 units

The total distance can be calculated using the formula for the sum of a geometric series:

Sn=a1−rn1−rS_n = a \frac{1 - r^n}{1 - r}

where a=4a = 4 (the first term) and r=12r = \frac{1}{2} (the common ratio).

Here, we need to find nn such that:

Sn=7.9375S_n = 7.9375

Calculating the total distance for each stage, we find that n=7n = 7 stages is when the total distance approximates 7.9375 units.

Step 2

Find the maximum distance the point can move, along its path, if it continues in this pattern indefinitely.

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Answer

The maximum distance can be found by calculating the sum of an infinite geometric series:

S∞=a1−rS_\infty = \frac{a}{1 - r}

Substituting a=4a = 4 and r=12r = \frac{1}{2}:

S∞=41−12=412=8S_\infty = \frac{4}{1 - \frac{1}{2}} = \frac{4}{\frac{1}{2}} = 8

Thus, the maximum distance the point can move if it continues indefinitely would be 8 units.

Step 3

Complete the second row of the table below showing the changes to the x co-ordinate.

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Answer

To find the changes in the x-coordinate for each stage:

  • Stage 1: +4+4 units
  • Stage 2: 00 units
  • Stage 3: −1-1 units
  • Stage 4: −0.5-0.5 units
  • Stage 5: −0.25-0.25 units
  • Stage 6: −0.125-0.125 units
  • Stage 7: −0.0625-0.0625 units
  • Stage 8: −0.03125-0.03125 units
  • Stage 9: −0.015625-0.015625 units

Thus, for the changes provided:

Stage123456789
Chg x+40-1-0.5-0.25-0.125-0.0625-0.03125-0.015625

Substituting these changes into the table correctly gives us the x-coordinate as 16/516/5 or (3,2.6)(3, 2.6) for the final position.

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