Walkabout wrote:
The length of the edging that surrounds circular garden K is 1/2 the length of the edging that surrounds circular garden G. What is the area of garden K (Assume that the edging has negligible width.)
(1) The area of G is \(25\pi\) square meters.
(2) The edging around G is \(10\pi\) meters long.
Solution:
We are given that the length of edging that surrounds circular garden K is ½ the length of the edging that surrounds circular garden G. Since the gardens are circular, we know that the circumference of garden K is ½ the circumference of circular garden G. We will use the circumference formula C = 2∏r. If we let G = the radius of garden G, and K = the radius of garden K, we can create the following equation.
2∏K = ½(2∏G)
We need to determine the area of garden K, using the area formula A = ∏r^2. Since K = the radius of garden K, we know:
Area of garden K = ∏K^2
Thus, if we can determine the value of K, we can determine the area of garden K.
Statement One Alone:The area of G is 25∏ square meters.
We can use the information in statement one to determine the value of the radius of garden G.
25∏ = ∏G^2
25 = G^2
√25 = √G^2
5 = G
Since we have the value of G, we can determine the circumference of garden G.
circumference of garden G = 2∏G
circumference = 2∏ x 5
circumference = 10∏
From the given information we also know that:
2∏K = ½(2∏G)
Since 10∏ is the circumference of garden G, we can substitute 10∏ for 2∏G in the equation 2∏K = ½(2∏G). We can now determine a value for K.
2∏K = ½(10∏)
2∏K = 5∏
K = 2.5
Since the radius of garden K is 2.5, we have enough information to determine the area of garden K. Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.
Statement Two Alone:The edging around G is 10∏ meters long.
Using the information in statement two, we know that the circumference of garden G is 10∏.
Recall that in statement one, we already determined that the circumference of garden G is 10∏. This is enough information to determine the radius of garden K and hence the area of garden K. Statement two alone is also sufficient to answer the question.
The answer is D.