Walkabout wrote:
Attachment:
Circle.png
The figure above shows a circular flower bed, with its center at O, surrounded by a circular path that is 3 feet wide. What is the area of the path, in square feet?
(A) \(25\pi\)
(B) \(38\pi\)
(C) \(55\pi\)
(D) \(57\pi\)
(E) \(64\pi\)
In solving this problem we first must recognize that the flower bed is the right triangle with sides of y yards, x yards, and z yards. We are given that the area of the bed (which is the right triangle) is 24 square yards. Since we know that area of a triangle is ½ Base x Height, we can say:
24 = ½(xy)
48 = xy
We also know that x = y + 2, so substituting in y + 2 for x in the area equation we have:
48 = (y+2)y
48 = y^2 + 2y
y^2 + 2y – 48 = 0
(y + 8)(y – 6) = 0
y = -8 or y = 6
Since we cannot have a negative length, y = 6.
We can use the value for y to calculate the value of x.
x = y + 2
x = 6 + 2
x = 8
We can see that 6 and 8 represent two legs of the right triangle, and now we need to determine the length of z, which is the hypotenuse. Knowing that the length of one leg is 6 and the other leg is 8, we know that we have a 6-8-10 right triangle. Thus, the length of z is 10 yards.
If you didn't recognize that 6, 8, and 10 are the sides and hypotenuse of a right triangle, you would have to use the Pythagorean to find the length of the hypotenuse: 6^2 + 8^2 = c^2 → 36 + 64 = c^2 → 100 = c^2. The positive square root of 100 is 10, and thus the value of z is 10.
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