Problem Solving

A cow is tethered to the corner of a rectangular 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? (The cow cannot enter the shed).

Nifty question.

You essentially have three circles with different radii.

One part has a radius of 5 - the length of the tether and this circle stretches through 270 degrees (3/4 area).

When it runs along the edge of a wall - at its full length it can wrap around the wall providing two small circles with radii of 1 and 2 that stretch for 90 degrees (1/4 area) - see below.



Ok so now we can break it down into three parts.

I have 3 quarters of the radius 5 circle = 0.75 x π x (5x5) = 18.75π

1 quarter of the radius 2 circle = 0.25 x π x (2x2) = π

1 quarter of the radius 1 circle = 0.25 x π x (1x1) = 0.25π

Therefore in total we have 20Ï€.

As the cow moves around the shed, the rope tethering the cow will WRAP AROUND the shed, as shown in the figure above.
As the cow moves COUNTERCLOCKWISE, the 5-unit rope will wrap 4 UNITS TO THE LEFT (along the top of the shed) and 1 UNIT DOWN (along the left side of the shed).
As the cow moves CLOCKWISE, the 5-unit rope will wrap 3 UNITS DOWN (along the right side of the shed) and 2 UNITS TO THE LEFT (along the base of the shed).
The result is that the cow can access the 3 shaded regions.

Blue region:
This region is equal to 3/4 of a circle with a center at A and a radius of 5.
Area = (3/4)(π)(5²) = (75/4)π.

Yellow region:
This region is equal to 1/4 of a circle with a center at B and a radius of 2.
Area = (1/4)(π)(2²) = π.

Red region:
This region is equal to 1/4 of a circle with a center at C and a radius of 1.
Area = (1/4)(π)(1²) = (1/4)π.

Total accessible area = blue region + red region + yellow region = (75/4)π + (1/4)π + π = (76/4)π + π = 20π.

The correct answer is E.