SOLUTION: A goat is tethered to the side of a square 10ft x 10ft shed. The tethering rope is 18 feet long and tied to a post in the middle of one of the sides. What ist he total area availab

Algebra ->  Formulas -> SOLUTION: A goat is tethered to the side of a square 10ft x 10ft shed. The tethering rope is 18 feet long and tied to a post in the middle of one of the sides. What ist he total area availab      Log On


   

Question 217114: A goat is tethered to the side of a square 10ft x 10ft shed. The tethering rope is 18 feet long and tied to a post in the middle of one of the sides. What ist he total area available for the goat to graze? Assume the shed is in the middle of a 100ft x 100ft field.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Step 1) Draw a 10' x 10' square to represent the shed.




Step 2) Extend a line segment from the midpoint of one side of the square (I chose the top) 18 ft out. This is the goat's leash.



Step 3) Now extend a segment 18 ft to the left (you could go either way). Take note how there's 5 feet from the midpoint to the corner.



Step 4) Sweep the two 18' segments (or just one) to form the segment shown in blue below. We'll call this area A%5B1%5D




Step 5) Since the rope is eventually going to be caught on the shed, this means that we now have 13 feet (instead of 18') of rope. So just draw out an arbitrary 13' segment




Step 6) Sweep a circular segment through to get the red area shown below in which we'll call A%5B2%5D




Step 7) Since 10 ft of rope is stuck on the wall, we now have only 3 ft of rope left. So sweep out the remaining area that the goat can roam (for this side at least). We'll call this new area in green A%5B3%5D




Step 8) From the last drawing, we see that there are 3 quarter circles which have the radii 18, 13, and 3 feet.

Recall that the area of a circle is A=pi%2Ar%5E2. Divide this by 4 to find the area of a quarter circle (since all of these segments are quarter circles) to get A=%28pi%2Ar%5E2%29%2F4

Using the last formula, we then get the areas:

A%5B1%5D=%28pi%2A%2818%29%5E2%29%2F4=%28pi%2A324%29%2F4=81pi


A%5B2%5D=%28pi%2A%2813%29%5E2%29%2F4=%28169pi%29%2F4


A%5B3%5D=%28pi%2A%283%29%5E2%29%2F4=%289pi%29%2F4



Now add the three individual areas: A%5B1%5D%2BA%5B2%5D%2BA%5B3%5D=81pi%2B%28169pi%29%2F4%2B%289pi%29%2F4=%28251pi%29%2F2

So the area that the goat can graze on the entire left side is %28251pi%29%2F2 square feet.

To find the total area, simply multiply that last value by 2 (since the right side is a reflection of the left side) to get 2%28%28251pi%29%2F2%29=%28cross%282%29%2A251pi%29%2Fcross%282%29=251pi

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Answer:
So the total area that the goat can graze is 251pi square feet. If we use the approximation 3.14 for pi , then the approximate area is 251%2A3.14=788.14 square feet.

So the total area available for the goat to graze is 788.14 square feet.

Note: if you need better precision, then just use a better approximation of pi.