Peanutís Backyard

By:Allison K. Kootsikas


 

Peanut, the overexcited dog, has lots of energy and is tearing apart the house. Peanut's owner decides to tie Peanut up in the backyard. He puts a stake in the middle of the yard and attaches one end of a rope to the stake and the other end to Peanut's collar.

Still full of energy, Peanut runs in circles, stretching the rope as far as he can from the stake. He makes 40 laps around the stake and travels a total distance of 1 mile. All this running has left Peanut tired and thirsty. If his bowl of water is 20 feet away from the stake, can he get a drink? Explain your answer. (Assume the rope is tied around the stake with a loop so that it will not wrap around the stake.)


(Source: The Math Forum)

   Extensions


What area does Peanut have to play in outside, when he is tied to his stake?


I decided to tackle this problem algebraically.Because Peanut was running around the outside of the circle, I decided to find the circumference of the circle.Once I find the circumference of the circle, I will need to find the radius of the circle, which will be the length of Peanutís rope.

 

One lap is 1/40th of a mile because it took 40 laps to equal one mile.

 

The formula for the circumference of a circle is C=2Pr

I divided 5,280 by 40 to find the distance of one lap, which would give me the circumference.

 

5,280/40=132 ft.

C=2Pr=132 ft.

 

Now that I have the circumference, I need to work backwards to find the radius, which will give us the length of Peanutís rope.

 
r=132/(2P)=66
66/P=21.02 ft.
 
Peanutís rope is 21 ft. long.He will have enough rope to get a drink of water, with one foot remaining.



Extension:What area does Peanut have to play in outside, when he is tied to the stake?
 
The formula for the area of a circle is A=Pr2
 
We know that the radius is 21 ft. from the earlier problem.
 
A=P(212)
A=P(441)
A=1,384.74 ft2
 
Peanut has 1,384.74 sq. ft. to play in.