How many unique ways are there to arrange the letters in the word HATTER?
2 answers:
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Refer to the diagram below. We need to find the areas of the green and blue regions, then subtract to get the area of the orange track only.
The larger green region is composed of a rectangle of dimensions 200 meters by 4+42+4 = 50 meters, along with two semicircles that combine to make a full circle. This circle has radius 25 meters.
The green rectangle has area 200*50 = 10000 square meters. The green semicircles combine to form an area of pi*r^2 = pi*25^2 = 625pi square meters. In total, the full green area is 10000+625pi square meters. I'm leaving things in terms of pi for now. The approximation will come later.
The blue area is the same story, but smaller dimensions. The blue rectangle has dimensions 200 meters by 42 meters, so its area is 200*42 = 8400 square meters. The blue semicircular pieces combine to a circle with area pi*r^2 = pi*21^2 = 441pi square meters. In total, the blue region has area 8400+441pi square meters.
After we figure out the green and blue areas, we subtract to get the orange region's area, which is the area of the track only.
orange area = (green) - (blue)
track area = (10000+625pi) - (8400+441pi)
track area = 10000+625pi - 8400-441pi
track area = (10000-8400) + (625pi - 441pi)
track area = 184pi + 1600 is the exact area in terms of pi
track area = 2178.05304826052 is the approximate area when you use the pi constant built into your calculator. If you use pi = 3.14 instead, then you'll get 2177.76 as the approximate answer. I think its better to use the more accurate version of pi. Of course, be sure to listen/follow your teachers instructions.
