Geometry

[Deepthi Subbu]

The figure above shows the top side of a circular medallioin made of a circular piece of glass surrounded by a metal frame , represented by the shaded region . If the radius of the medallion is r centimeters and the width of the medal frame is s centimeters , then in terms of s and r , what is the area of the metal frame , in square centimeters?

1. pi (r-s)^2
2.pi(r^2 - s^2)
3.2pi( r-s)
4.r pie(2r - s)
5.pie s ( 2r - s)

[GMATGuruNY]

The figure above shows the top side of a circular medallioin made of a circular piece of glass surrounded by a metal frame , represented by the shaded region . If the radius of the medallion is r centimeters and the width of the medal frame is s centimeters , then in terms of s and r , what is the area of the metal frame , in square centimeters?

1. pi (r-s)^2
2.pi(r^2 - s^2)
3.2pi( r-s)
4.r pie(2r - s)
5.pie s ( 2r - s)

We can plug in values for r and s.
Let r=3 and s=1.
Area of medallion = pi(3^2) = 9pi.
Radius of circular glass = radius of medallion - width of frame = 3-1 = 2.
Area of circular glass = pi(2^2) = 4pi.
Area of frame = medallion - glass = 9pi-4pi = 5pi. This is our target answer.

Now we plug r=3 and s=1 into all the answers to see which yields our target of 5pi.

Only answer choice E works:
Pi*s*(2r-s) = pi*1*(2*3 - 1) = 5pi.

The correct answer is E.

[Deepthi Subbu]

Thanks for the explanation , but there's one more thing that I want to ask .

Were the numbers that you picked are just random and will picking numbers work in all cases for such problems?

Cuz I tried solving it directly and it dint work out.

[GMATGuruNY]

Thanks for the explanation , but there's one more thing that I want to ask .

Were the numbers that you picked are just random and will picking numbers work in all cases for such problems?

Cuz I tried solving it directly and it dint work out.

Generally, when there are variables in the answer choices, we can pick values. The goal is to pick values that make the math easy.

[gmatmachoman]

Cuz I tried solving it directly and it dint work out.

Radius of Medallion : r = radius of circular piece of glass + radius of metal frame

Radius of circular piece of glass = r -s

Area of Medallio = Area of circular piece of glass + Area of Metal frame
Pi r ^2 = Pi ( r-s) ^2 + Area of Metal frame

Solving we get ,

Area of Metal frame = Pi s ( 2r - s)

Pick e

[Deepthi Subbu]

Thanks , this was what I was asking for.