A circular target is divided into nine parts of equal area by eight concentric rings. If the radius of the target is 27 cm, find the radius of the inner circle.
What is the ratio of the area of the inner circle to the area of the whole target?
What is the ratio of the dimensions of the inner circle to the dimensions of the whole target?
So the whole circle has an area of 27^2π = 729π
which means that each part has an area of 81π cm^2
The radius of the inner circle must be 9 cm -----> (9^2π = 81π)
the area of the 2nd circle must be 2(81π) = 162π , inner circle + first ring
πr2^2 = 162π
r2 = √162 = 9√2
the area of the third circle must be 3(81π) = 243π , inner circle + 2 rings
πr3^2 = 243π
r3 = √243 = 9√3
...
the area of the 7th circle = 7(81π) = 567π
r7^2π = 567π
r7 = √567 = 9√7
ahhh, nice pattern
so the radius of the 9th circle = 9√9 = 27 , as given
What is the ratio of the dimensions of the inner circle to the dimensions of the whole target
---> 9 : 27 = 1:3
What is the ratio of the area of the inner circle to the area of the whole target?
----> 81π : 729π = 1 : 9
I expected this result!, It is one of the properties of areas of similar shapes
Wow, thank you! I now better understand this topic. I think I'm ready for my test. Thanks!
To find the radius of the inner circle, we need to determine the radius difference between each pair of concentric rings.
Since the target is divided into nine equal parts by eight concentric rings, we have eight rings, indicating there are nine sections in total.
To find the radius difference between each pair of concentric rings, we divide the total radius (27 cm) by the number of sections (9):
Radius difference = Total radius / Number of sections
Radius difference = 27 cm / 9
Radius difference = 3 cm
This means that the radius of the inner circle is equal to the total radius minus the radius difference:
Radius of the inner circle = Total radius - Radius difference
Radius of the inner circle = 27 cm - 3 cm
Radius of the inner circle = 24 cm
Therefore, the radius of the inner circle is 24 cm.
Now, let's calculate the ratio of the area of the inner circle to the area of the whole target.
The area of a circle is given by the formula: A = πr^2
Area of the inner circle = π * (Radius of the inner circle)^2 = π * 24^2
Area of the inner circle = 576π cm^2
Area of the whole target = π * (Total radius)^2 = π * 27^2
Area of the whole target = 729π cm^2
The ratio of the area of the inner circle to the area of the whole target is:
Ratio = Area of the inner circle / Area of the whole target
Ratio = 576π cm^2 / 729π cm^2
Ratio = 576 / 729
Ratio = 16 / 27
Therefore, the ratio of the area of the inner circle to the area of the whole target is 16:27.
To find the ratio of the dimensions of the inner circle to the dimensions of the whole target, simply divide the radius of the inner circle by the total radius:
Ratio = Radius of the inner circle / Total radius
Ratio = 24 cm / 27 cm
Ratio = 8 / 9
Therefore, the ratio of the dimensions of the inner circle to the dimensions of the whole target is 8:9.
To find the radius of the inner circle, we can use the fact that the target is divided into nine equal parts. Since there are eight concentric rings, we can determine that each ring corresponds to 1/9th of the total area of the target.
To find the area of each ring, we first calculate the area of the whole target using the formula for the area of a circle: A = πr^2, where A is the area and r is the radius. In this case, the radius of the target is given as 27 cm. Therefore, the area of the whole target is:
A_target = π(27 cm)^2
To determine the area of each ring, we can divide the area of the whole target by the number of rings:
A_ring = A_target / 8
Since each ring corresponds to 1/9th of the total area, the area of the inner circle is:
A_inner_circle = A_ring * 1/9
Now, we can calculate the radius of the inner circle. The area of a circle is given by the formula mentioned earlier, so we can rearrange the formula to solve for the radius:
A_inner_circle = πr^2
r_inner_circle = √(A_inner_circle / π)
Substituting the value of A_inner_circle:
r_inner_circle = √((A_ring * 1/9) / π)
Now, we can substitute the value of A_ring:
r_inner_circle = √((A_target / 8) * 1/9 / π)
r_inner_circle = √((π(27 cm)^2 / 8) * 1/9 / π)
Simplifying the expression:
r_inner_circle = √(27 cm^2 / 8 / 9)
r_inner_circle = √(27 cm^2 / 72)
r_inner_circle = √(3/8) * 27 cm
r_inner_circle = (√3 / 2) * 27 cm
Therefore, the radius of the inner circle is (√3 / 2) * 27 cm.
To find the ratio of the area of the inner circle to the area of the whole target, we can use the formula for the area of a circle:
Area_ratio = A_inner_circle / A_target
Area_ratio = (A_ring * 1/9) / A_target
Substituting the values:
Area_ratio = A_ring / (9 * A_target)
Area_ratio = (A_target / 8) / (9 * A_target)
Canceling out the A_target:
Area_ratio = 1 / (8 * 9)
Area_ratio = 1 / 72
Therefore, the ratio of the area of the inner circle to the area of the whole target is 1:72.
To find the ratio of the dimensions (i.e., radius or diameter) of the inner circle to the dimensions of the whole target, we can use the formula for the area of a circle:
Dimension_ratio = r_inner_circle / r_target
Substituting the values:
Dimension_ratio = (√3 / 2) * 27 cm / 27 cm
Canceling out the common factors:
Dimension_ratio = √3 / 2
Therefore, the ratio of the dimensions of the inner circle to the dimensions of the whole target is √3:2.