A pen is the shape of an equilateral triangle. A goat is attached to a corner of the pen on a rope. The goat eats all the grass it can reach. It can just reach the opposite fence of the pen. What percentage of the grass in the pen does the goat eat?
- Step 1: Area of the penA triangle’s area is base * height / 2. Let one side (the base) have length x. By splitting the triangle in half and using Pythagoras’s rule, we get x2 = (x/2)2 + h2. This rearranges to h = sqrt( x2 – (x/2)2 ) = sqrt( 3 * x2 / 4 ) = sqrt(3) * x / 2.Now, we can create a formula for the area, using the base and height. The area = sqrt(3) * x2 / 4
- Step 2: Area of the segmentSince the goat just reaches the opposite wall, the length of rope must be equal to the traingle’s height, which we calculated above: sqrt(3) * x / 2. This is the radius of the segment. The area of the circle is pi * radius2. However, the goat cannot reach a full circle. The interior angle of an equilateral triangle is 60o = pi/3 c. The goat can access 60/360 of the circle. So, the area of the segment is 60/360 * pi * (sqrt(3) * x / 2)2 = pi/6 * 3/4 * x2 = x2 * pi/8
- Step 3: Finding the percentageWe take the ratio of the segment area to the triangle area. That is, x2 * pi/8 divided by sqrt(3) * x2 / 4, which is pi / (sqrt(3) * 2) = 90.7% (3.s.f)
MC
