Asked by rem
A rectangular pool has side lengths of 12 yards and 6 yards. An octagon is drawn around the pool. The octagon has 2 sides with lengths 12 yards, and 6 sides with lengths 6 yards. The area between the rectangle and the octagon is shaded.
How many square yards of cement are needed to create the walkway around the rectangular pool?
All Answers 1
Answered by
GPT-5 mini
AI
Let the four corner cutoffs be isosceles right triangles with leg length x. Then each diagonal side of the octagon has length x√2, and we are told those are 6, so x√2 = 6 ⇒ x = 3√2.
The outer rectangle (before cutting corners) has dimensions
12 + 2x = 12 + 6√2 and 6 + 2x = 6 + 6√2.
Its area is (12 + 6√2)(6 + 6√2) = 144 + 108√2.
Remove the four corner triangles (total area 4·(1/2)x^2 = 2x^2 = 36) to get the octagon area:
octagon area = 144 + 108√2 − 36 = 108 + 108√2.
The pool area is 12·6 = 72, so the walkway area is
(108 + 108√2) − 72 = 36 + 108√2 = 36(1 + 3√2) square yards.
Answer: 36 + 108√2 square yards.
The outer rectangle (before cutting corners) has dimensions
12 + 2x = 12 + 6√2 and 6 + 2x = 6 + 6√2.
Its area is (12 + 6√2)(6 + 6√2) = 144 + 108√2.
Remove the four corner triangles (total area 4·(1/2)x^2 = 2x^2 = 36) to get the octagon area:
octagon area = 144 + 108√2 − 36 = 108 + 108√2.
The pool area is 12·6 = 72, so the walkway area is
(108 + 108√2) − 72 = 36 + 108√2 = 36(1 + 3√2) square yards.
Answer: 36 + 108√2 square yards.
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