A two dimensional picture of a circular onion ring is shown at the right. The inner radius of the onion ring is x cm. and the thickness of the ring is 1 cm. Which of the numbered choices represents the area, in square centimeters, of the onion ring (the shaded region)?
1) (2x+1)*pi
2) 2*pi
3) 2x*pi
4) pi
Since you can't show the diagram, I will surmise the following
inner radius --- x
outer radius --- x+1
area of onion ring
= π(x+1)^2 - x^2)
= π(x^2 + 2x + 1 - x^2)
= π(2x+1) , looks like the first choice
To find the area of the onion ring, we need to subtract the area of the inner circle from the area of the outer circle.
The inner circle has a radius of x cm, so its area is given by πx^2.
The outer circle has a radius of (x + 1) cm (since the thickness of the ring is 1 cm). So its area is given by π(x + 1)^2.
To find the area of the onion ring, we subtract the area of the inner circle from the area of the outer circle:
π(x + 1)^2 - πx^2
Expanding and simplifying this expression, we get:
π(x^2 + 2x + 1) - πx^2
This simplifies to:
πx^2 + 2πx + π - πx^2
The πx^2 and -πx^2 cancel each other out, leaving us with:
2πx + π
Now, we can factor out π from both terms:
π(2x + 1)
Therefore, the area of the onion ring (shaded region) is represented by choice 1) (2x + 1)π.
To find the area of the onion ring, we need to subtract the area of the inner circle from the area of the outer circle. The inner circle has a radius of x cm and the outer circle has a radius of x + 1 cm (considering the thickness of the ring is 1 cm).
The formula to find the area of a circle is A = πr^2, where A is the area and r is the radius.
The area of the inner circle is A_inner = π(x)^2 = πx^2.
The area of the outer circle is A_outer = π(x + 1)^2 = π(x^2 + 2x + 1) = πx^2 + 2πx + π.
Now, to find the area of the onion ring, we subtract the area of the inner circle from the area of the outer circle:
A_ring = A_outer - A_inner = (πx^2 + 2πx + π) - πx^2 = 2πx + π.
So, the correct answer choice is 2πx + π, which corresponds to option 3).