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?
What is the radius of the inner circle, in cm?
I usually understand math, but today, I'm having some trouble.
Could you elaborate a little more, I still don't understand.
Are you in SOM (Spiritofmath)
Ah, math can be a tricky circus sometimes, but don't worry, I'm here to clown around and help you out! Let's get to it!
To find the radius of the inner circle, we need to figure out how much each concentric ring contributes to the total area of the target.
Now, since we have eight concentric rings, we can divide the target into nine parts of equal area. This means that each of those parts is equal to the area of one ring.
To find the area of the whole target, we can use the formula for the area of a circle: A = πr^2. Since the radius of the target is given as 27 cm, we can substitute it into the formula: A_target = π(27^2) = 729π cm^2.
Since each ring has equal area, we can divide the total area of the target by 9 to find the area of one ring: A_ring = A_target / 9 = (729π) / 9 = 81π cm^2.
Now, let's find the radius of the inner circle. The inner circle corresponds to the first and smallest concentric ring. Let's call its radius "r".
The area of the first ring is equal to πr^2. We know that this area is 81π cm^2. So, we can set up the following equation:
πr^2 = 81π
Simplifying, we find:
r^2 = 81
Taking the square root of both sides, we get:
r = 9
Therefore, the radius of the inner circle is 9 cm.
Now, let's move on to the ratios!
The ratio of the area of the inner circle to the area of the whole target is:
(A_inner circle) / (A_target) = (π9^2) / (π27^2) = (π * 81) / (π * 729) = 81/729 = 1/9
In other words, the area of the inner circle is one-ninth the area of the whole target.
Next, the ratio of the dimensions of the inner circle to the dimensions of the whole target is simply the ratio of their radii:
(r_inner circle) / (r_target) = 9 / 27 = 1/3
So, the dimensions of the inner circle are one-third the dimensions of the whole target.
Hope that clarifies things for you! Remember, math can be a clownishly fun challenge sometimes.
No problem! Let's break down each question step by step.
To find the radius of the inner circle, we need to consider that the circular target is divided into nine equal parts by eight concentric rings. This means that there are nine areas, including the inner circle. Since the target's area is divided equally among these nine parts, each segment occupies 1/9th of the total area of the target.
To determine the radius of the inner circle, we'll need to set up the equation as follows:
Area of the target = 9 * Area of the inner circle
The formula for the area of a circle is A = πr^2, where 'A' represents the area and 'r' denotes the radius. Substituting the given values, we get:
π * (27^2) = 9 * π * (r^2)
729π = 9π * r^2
Canceling out the π terms on both sides, we get:
r^2 = 729 / 9
r^2 = 81
Taking the square root of both sides, we find:
r = 9
So, the radius of the inner circle is 9 cm.
Moving on to the next question, let's find the ratio of the area of the inner circle to the area of the whole target. We can use the formulas mentioned earlier to calculate these values.
The ratio of the area of the inner circle to the area of the whole target can be expressed as:
(Area of the inner circle) / (Area of the target) = (π * r^2) / (π * (27^2))
Canceling out the π terms, we have:
(r^2) / (27^2) = (9^2) / (27^2) = 1 / 3
So, the ratio of the area of the inner circle to the area of the whole target is 1:3.
Finally, let's determine the ratio of the dimensions of the inner circle to the dimensions of the whole target. Since the dimensions of a circle are directly related to the radius (i.e., twice the radius gives the diameter), we can calculate these ratios based on the radius values.
The ratio of the radius of the inner circle to the radius of the target is:
9 / 27 = 1 / 3
Thus, the ratio of the dimensions of the inner circle to the dimensions of the whole target is 1:3.
I hope this helps! Let me know if there's anything else you need assistance with.
The 8 circles divide the target into 9 parts, so you want the area of each ring to be 729π/9 = 81π
So, the inner circle must have radius 9, since its area must also be 81π
You should be able to answer the questions now