Tough math question?

Question

A goat is tied to the outside corner of 4 meter by 4 meter pen on a 16 meter leash. How many square meters of grass can he eat.
I thought this was going to be easy but it has me stumped. Good luck.

Answer 1

This IS a good problem. I just spent a few minutes with it and have the correct answer:

The area outside the pen accesible to the goat is bounded by 3 arcs, one of 16 m radius, the other 2 of 12 m radius. The 2 12 m radii each swing through an arc of 59.16124 deg at their point of intersection. The remaining area is composed of 2 triangles with a common side and apex where the 12 m circles intersect. The sides of the triangles can be determined fairly easily and thus their areas.

Area of 3/4 circle of 16 m radius = .75*pi*16^2 = 603.185
Area of 2 12m circles (angle above) = (118.322/360)*pi*12^2 = 148.688
Area of triangles = 24.98483

The summation of which is 776.858 m^2

Answer 2

First, I believe you'll take 75% of the area of a circle with a radius of 16 meters. The goat can travel unrestricted for 3/4 of a circle.

To that, add 25% of the area of a circle with a radius of 12 meters from which you've subtracted 16 square meters. Once he turns either corner of the pen, the effective length of his leash is reduced to 12 meters. And I'm assuming he can't get INTO the pen to eat the grass within.

Looks like 700.1936 square meters, dependent upon to what decimal point you use PI.

Answer 3

If he is inside the pen, he can eat 16 sq meters of grass. If he is outside the pen and cannot access grass inside, then you have to compute area of 32 meter circle, less the 16 meters inside the pen. PiRsquared is formula for area of a circle, so 16 x pi squared less the 16 square meters inside the pen will be amount he can eat. Approx 2510 meters. (Didn't leave decimals)

Answer 4

Man, you're right - this one is tough. I spent my lunch period trying to break this into pieces I could calculate, but it's too much for my feeble mind to figure out in 30 minutes. The best I could do is figure out the goat can almost do a full circle, minus the part inside the fence so the answer should be a little less than 256*(pi) - 16 = 788.25. (The part the goat can't reach that is inside the circle but outside the fence is long but very skinny.) Since Steve's answer is a little less than 788.25, I believe him.

Answer 5

if he is on hte inside, 16 square meters i think, because 4 x 4 is 16, but if outside i have no idea

Answer 6

pi x 16^2 squ.mts.