Larry The Lamb

Geometry Level 2

A square field $ABCD$ has center $O$ and side length $2l$ . A rope of length $l$ connects Larry the lamb at one end to $O$ at the other end.

The perpendicular from $O$ to side $AD$ forms a wall that Larry cannot pass through. The point $P$ that is the midpoint of the perpendicular from $O$ to $DC$ forms a peg that will catch on the rope.

What percentage of the field can Larry walk on, to the nearest percent?

100% 63% 50% 49% 79%

3 solutions

Maria Kozlowska
Feb 12, 2016

He can walk area of a circle with radius $l$ . Peg and wall do not limit the area, he can approach the wall from both sides, and he can walk between wall and a peg to finish the full circle. $\frac{\pi * l^2}{4*l^2} \approx 0.79$

Why is the denominator term $4*l^2$

Ajit Deshpande - 5 years, 3 months ago

It is area of the square.

Maria Kozlowska - 5 years, 3 months ago

Lucas Erick
Feb 17, 2016

Larry can describe an entire circle with a L radius. The peg will not restrain him, because it can pass between the wall and the peg. Therefore: pi.L^2/4L^2=3,14/4=0,79

Ashish Menon
Feb 16, 2016

within the black margin the lamb has no restrictionfor walking on the grass. But, the peg and the wall does not allow it to walk 100% of the field. The answer is somewhere between 75 and 100% and 79% is the only option which has the mentioned characteristics. After the rope gets attached to the peg, the lamb can walk some more distance.

This solution is not entirely correct, Larry cannot reach vertex B. Why not? Also, can you explain what part of the region outside of the black margin Larry can reach?

Michael Fuller - 5 years, 3 months ago

I kbow, it is just an assumption. If you want me to delete the answer i would.

Ashish Menon - 5 years, 3 months ago

No need of deleting your answer. It is just an approximation . Since The answer should be in between 100% and 50% your approach in this instance is correct.

Gunawardana Gamagedara - 5 years, 3 months ago

@Gunawardana Gamagedara – Thank you,sir

Ashish Menon - 5 years, 3 months ago

Larry The Lamb

3 solutions

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