SOLUTION: a goat is tied to a corner of a 30 ft by 35 ft building. if the rope is 40 ft long and the goat can reach 1 ft farther than the rope length, what is the maximum area the goat can c

Algebra ->  Surface-area -> SOLUTION: a goat is tied to a corner of a 30 ft by 35 ft building. if the rope is 40 ft long and the goat can reach 1 ft farther than the rope length, what is the maximum area the goat can c      Log On


   

Question 1178371: a goat is tied to a corner of a 30 ft by 35 ft building. if the rope is 40 ft long and the goat can reach 1 ft farther than the rope length, what is the maximum area the goat can cover?
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

IMG-0483

Given:
width of building = 30 ft
length of building = 35 ft
length of rope = 40 ft
total reach of goat = 41 ft
To solve for the total area, get the sum of the three areas.
A%5Bt%5D=A%5B1%5D%2BA%5B2%5D%2BA%5B3%5D
Area 1 is equal to the one quarter of the area of a 6+ft radius circle.
A%5B1%5D=%281%2F4%29pi%2Ar%5E2
A%5B1%5D=%281%2F4%29pi%2A6%5E2
A%5B1%5D=28.274
Area 2 is equals to the one quarter of the area of a 11 ft radius circle.
A%5B2%5D=%281%2F4%29pi%2A11%5E2%0D%0A%0D%0A%7B%7B%7BA%5B2%5D=95.03
Area 3 is equals to the three quarters of the area of a 41 ft radius circle.
A%5B3%5D=%283%2F4%29pi%2A41%5E2%0D%0A%0D%0A%7B%7B%7BA%5B3%5D=3960.76
total area:
A%5Bt%5D=28.274%2B95.03%2B3960.76
A%5Bt%5D=4084


Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.

The accessible area is the disjoint union of 5 (five, FIVE) 90° circular sectors



A%5B1%5D = %28pi%2A41%5E2%29%2F4


A%5B2%5D = %28pi%2A41%5E2%29%2F4


A%5B3%5D = %28pi%2A41%5E2%29%2F4


A%5B4%5D = %28pi%2A11%5E2%29%2F4


A%5B5%5D = %28pi%2A6%5E2%29%2F4.


Therefore, the total accessible area is


    A = A%5B1%5D%2BA%5B2%5D%2BA%5B3%5D%2BA%5B4%5D%2BA%5B5%5D = %283.14159%2F4%29%2A%2841%5E2%2B41%5E2%2B41%5E2%2B11%5E2%2B6%5E2%29 = 4084.07 square feet   (rounded).      ANSWER

Solved.