the diameter of two circles are in the ratio of 2:3. Find the area of the smaller circle if the area of the larger circle is 63pie square meters
the areas are in the ratio 4:9, so
4/9 * 63pi = ?
Note that's pi, not pie!
Steve, can you elaborate on that?
To find the area of the smaller circle, we need to know the ratio of their diameters.
Let's assume the diameter of the smaller circle is 2x, and the diameter of the larger circle is 3x.
The formula for the area of a circle is A = πr^2, where r is the radius of the circle.
Since the diameter is twice the radius, we can write:
For the smaller circle:
Radius (r) of the smaller circle = (2x)/2 = x
For the larger circle:
Radius (r) of the larger circle = (3x)/2 = (3/2)x
Given that the area of the larger circle is 63π square meters, we can set up the equation:
π((3/2)x)^2 = 63π
Simplifying:
(3/2)^2πx^2 = 63π
(9/4)πx^2 = 63π
Cancelling out π from both sides:
(9/4)x^2 = 63
Now, we can solve for x:
(9/4)x^2 = 63
Multiplying both sides by 4/9:
x^2 = (4/9) * 63
x^2 = 28
Taking square root on both sides:
x = √28
x ≈ 5.29
Now, we can find the area of the smaller circle:
Area of the smaller circle = πr^2
Area = π(x)^2 = π(5.29)^2 ≈ 88.16 square meters
Therefore, the area of the smaller circle is approximately 88.16 square meters.
To find the area of the smaller circle, we need to know the ratio between the diameters of the two circles. We are given that the ratio of the diameters is 2:3.
Let's assign a variable to represent the diameter of the smaller circle. We'll call it "d". Since the ratio of the diameters is 2:3, the diameter of the larger circle can be represented as 3d.
The formula to calculate the area of a circle is A = πr^2, where A is the area and r is the radius of the circle.
The radius of the smaller circle is half of its diameter, so it can be represented as r = d/2.
Similarly, the radius of the larger circle is half of its diameter, so it can be represented as R = (3d)/2.
We are given that the area of the larger circle is 63π square meters, so we can write the equation:
A_large = πR^2 = 63π.
Substituting the values, we get:
π((3d)/2)^2 = 63π.
Simplifying further:
(9d^2)/4 = 63.
Now, we can solve for d by multiplying both sides of the equation by 4/9:
d^2 = (63 * 4)/9.
d^2 = 28.
Taking the square root of both sides:
d = √28.
d ≈ 5.29 meters.
Now that we have the diameter of the smaller circle, we can calculate its radius:
r = d/2 = 5.29/2 = 2.65 meters.
Finally, we can calculate the area of the smaller circle using the formula:
A_small = πr^2 = π(2.65)^2.
A_small ≈ 22.01π square meters.
So, the area of the smaller circle is approximately 22.01π square meters.