The figure consists of two circles that have the same center. If the shaded area is 64 pi square inches and the smaller circle has stadiums of 6 inches, what is the radius, in inches, of the larger circle?
πr^2 - 36π = 64π
r^2 = 100
r = 10
Well, let me put on my math clown hat and calculate that for you! If the smaller circle has a radius of 6 inches, then its area can be calculated using the formula A = πr^2. Plugging in the radius, we get A = π(6^2) = 36π square inches.
Now, we know that the shaded area is 64π square inches. To find the area of the larger circle, we need to subtract the area of the smaller circle from the shaded area. So, the area of the larger circle is 64π - 36π = 28π square inches.
To find the radius of the larger circle, we can use the formula A = πr^2 again. Plugging in the area we just calculated, we get 28π = πr^2. Dividing both sides by π, we get 28 = r^2. Taking the square root of both sides, we find that the radius of the larger circle is √28, which is approximately 5.29 inches.
So, the radius of the larger circle is approximately 5.29 inches. Just remember, even clowns can do math!
To find the radius of the larger circle, we need to first determine the area of the shaded region.
Since the shaded region is given as 64π square inches, we can set up an equation:
Area of shaded region = Area of larger circle - Area of smaller circle
64π = πr₁² - πr₂²
Given that the radius of the smaller circle is 6 inches, we can substitute it into the equation:
64π = πr₁² - π(6)²
Now, we can simplify the equation:
64π = πr₁² - 36π
Rearranging the terms:
64π + 36π = πr₁²
Simplifying further:
100π = πr₁²
We can now cancel out the π from both sides of the equation:
100 = r₁²
Taking the square root of both sides:
√100 = r₁
Therefore, the radius of the larger circle is 10 inches.
To find the radius of the larger circle, we first need to know the radius of the smaller circle. From the problem, we are given that the radius of the smaller circle is 6 inches.
The shaded area is the difference between the area of the larger circle and the area of the smaller circle. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
Let's denote the radius of the larger circle as R.
The area of the larger circle is πR^2.
The area of the smaller circle is π(6)^2 = 36π.
The shaded area is then given by:
πR^2 - 36π = 64π square inches.
Simplifying the equation, we have:
πR^2 = 64π + 36π
πR^2 = 100π
R^2 = 100
R = √100
R = 10 inches
Therefore, the radius of the larger circle is 10 inches.