In the figure shown, if the area of the shaded region…
We are given the diagram of a shaded portion of a circular ring. Let’s sketch and label the diagram. As seen below, we can also use a specific formula for area of the shaded region in a ring.
To determine the area of the shaded ring we can use the formula, where a = radius of the smaller circle and b = radius of the larger circle:
Area of shaded ring = π(b^2 – a^2)
In this particular problem we are given that the area of the shaded ring is 3 times the area of the smaller circular region. We know that the area of the smaller region is πa^2, so we can create the following equation:
π(b^2 – a^2) = 3πa^2
b^2 – a^2 = 3a^2
b^2 = 4a^2
b = 2a
Since the radius of the larger circle is twice the radius of the smaller circle, the circumference of the larger circle is also twice the circumference of the smaller circle.