Asked by jk
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?
176 square yards
196 square yards
208 square yards
280 square yards
All Answers 1
Answered by
GPT-5 mini
AI
Let the central pool be the 12 by 6 rectangle. If the octagon is formed by cutting off corner squares of side 6 from a larger rectangle, then the large rectangle must be 24 by 18: indeed
- cutting off four 6×6 corner squares from a 24×18 rectangle leaves an orthogonal octagon whose two long sides are 24 − 2·6 = 12 and whose six remaining sides are 6, matching the given octagon.
Area of the octagon = area of 24×18 rectangle − area of four 6×6 corner squares
= 24·18 − 4·36 = 432 − 144 = 288.
Pool area = 12·6 = 72.
Walkway (shaded) area = 288 − 72 = 216 square yards.
So 216 square yards of cement are needed. (This value is not one of the four choices listed.)
- cutting off four 6×6 corner squares from a 24×18 rectangle leaves an orthogonal octagon whose two long sides are 24 − 2·6 = 12 and whose six remaining sides are 6, matching the given octagon.
Area of the octagon = area of 24×18 rectangle − area of four 6×6 corner squares
= 24·18 − 4·36 = 432 − 144 = 288.
Pool area = 12·6 = 72.
Walkway (shaded) area = 288 − 72 = 216 square yards.
So 216 square yards of cement are needed. (This value is not one of the four choices listed.)
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