SOLUTION: In a circular tent, the last row of seats are to be placed 75-feet away from a central pole that is 56-feet high. Ropes from the top of the pole are to be staked into the ground i

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Question 1135793: In a circular tent, the last row of seats are to be placed 75-feet away from a central pole that is 56-feet high. Ropes from the top of the pole are to be staked into the ground in such a way that they are 6-feet above ground where the last row of seats are to be placed. At what distance from the central pole should the ropes be staked in? Please round to one decimal place.
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Answer: 84 feet
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Explanation on how to get that answer:

Draw out the diagram. Technically this isn't needed but it's handy for visual learners (such as myself).

Point A = base of the pole
Point B = top of the pole
Point C = base of the seats in the last row
Point D = highest point just above point C (aka the ceiling over the last row)
Point E = location where the rope is staked into the ground

As you can see, we have two similar triangles: Triangle ABE and triangle CDE
They can be proven similar by the AA (angle angle) similarity theorem.

Based on the points mentioned, we can say these facts
  • AB = 56 = the height of the pole
  • AC = 75 = distance from the base of the pole to the base of the stands in the last row
  • CD = 6 = clearance height for people in the last row
  • CE = x = distance from the base of the stands in the last row to where the rope is staked in the ground


Furthermore, we know that
AE = AC + CE
AE = 75 + x
based on the segment addition postulate

Now use the idea that the triangles are similar to set up a proportion. Then cross multiply to solve for x
%28AB%29%2F%28CD%29+=+%28AE%29%2F%28CE%29

%2856%29%2F%286%29+=+%2875%2Bx%29%2F%28x%29 Substitution

56x+=+6%2875%2Bx%29 Cross multiply

56x+=+6%2875%29%2B6%28x%29 Distribute

56x+=+450%2B6x

56x-6x+=+450%2B6x-6x Subtract 6x from both sides

50x+=+450

%2850x%29%2F50+=+450%2F50 Divide both sides by 50

x+=+9

This means CE = 9 and
AE = AC + CE
AE = 75 + x
AE = 75 + 9
AE = 84
which is the distance we want: the distance from the base of the pole to where the rope is staked into the ground.