area that is accessible to the cow

Question

A cow is tethered to the corner of a shed. If the length of the rope is 5, and the shed has length 4 and width 3, what is the maximum area that is accessible to the cow?

12pi
15pi
16pi
18pi
20pi

bryantmichaels · Answer

The answer I come up with quickly is 20 pi.

75% of the 5 foot radius circle is accessible
plus
25% of a one foot radius circle is accessible
plus
25% of a two foot radius circle is accessible

this becomes

3/4 pi*25 + 1/4 pi*4 + 1/4 pi*1

which reduces to 
pi

which reduces to

pi*20

KarishmaB · Answer

Pathetic diagram but it will give you an idea...
The curve shows the region that is accessible to the cow... 3/4 th of the circle with radius 5 is directly accessible.
Then the rope turns at the corners and a quarter of the area of the circle with radius 1 and a quarter of the circle with radius 2 are available. (If this is confusing, try to turn a long piece of thread around a rectangular 3D object.)

Attachment:

Ques2.jpg [ 6.08 KiB | Viewed 2271 times ]

Area of the 3/4th circle with radius 5 = \((3/4)pi*5^2\)
Area of 1/4th circle with radius 1 = (1/4)pi*1
Area of 1/4th circle with radius 2 =\((1/4)pi*2^2\)

Total accessible area = (75/4)pi + (1/4)pi + (4/4)pi = 20pi

KarishmaB · Answer

Hey Bryant,
Sorry. Didn't realize you were posting the solution at the same time. 
Karishma

USCTrojan2006 · Answer

I got E, but in an extremely dirty fashion.

We know the area of the circle is 25pi. The cow has all access to 3/4 of this area or 18.75pi. Also take note that the cow has extra access to the short sides of the rectangular shed within the circle. Use pythagoreum theorem and you know that the diagonal of the shed is the radius of the circle (3, 4, 5 triangle).

Since 18.75pi is greater than 18pi, and that the cow has extra access, 20pi was the best answer.

However, if this question had 19pi, 20pi, and 21pi as answers, this definitely wouldn't have been the best approach.

KarishmaB · Answer

Dirty solutions are just innovative solutions when you don't know the cleanest method. Absolutely nothing wrong with them if you can make the best of the situation using them. 
But what was your logic of using pythagorean theorem?

USCTrojan2006 · Answer

I just imagined the cow having a radius of 5 walking distance. The question gave length 3 and 4 for the rectangle. I imagined one of the edges of the rectangle at the center of the circle, since the cow was attached to one of the corners. Using the theorem, one could get the 3, 4, 5 triangle by halving the rectangle diagonally, and conclude that one of the edges hits the circle.

area that is accessible to the cow

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