This is just a rough drawing of what is needed to be found.
Here's the equation:
A botanist has been studying tree rings for a certain species of tree. A cross-section of a tree near the ground is shown (see picture above).
The botanist observes that:
1. After one year of growth, the trunk of the tree near the ground is approximately a circle with a 1 inch radius and [2] each successsive year's growth creates a ring with the same area as the original circle.
What I do not understand is how to find the area of the second ring (the ring around the innermost circle), when all that is given is radius of the smallest ring.
And how, the area will be the same as the original circle - without having the same radius. Hence: each successsive year's growth creates a ring with the same area as the original circle. Is the confusing part.
The Area of the smallest ring is A= 3.14 because the radius squared, which is 1 multiplied by pi, is pi itself.
How would the second ring be obtained?
Help?
I'm simply reading this off of my sheet, truthfully I've been pondering over it for quite a while...
And if the original circle was removed, the second 'ring' would be a circle. Therefore the area of the second 'ring/circle' needs to be found. What I don't understand is how to go about doing that, given only the smallest ring's radius.
If the original circle is removed, the 2nd ring will still be a ring. The area of the second ring is not dependent of the first circle except that they are the same since that is what the question stated.
So is the question asking you to find the area of the 2nd ring?
Unless I understood this question wrong, which is kinda hard to do since the question said " creates a ring with the same area as the original circle", then the area of the 2nd ring will be the same as the original circle...
I understand that fact now. I simply assumed that since the radius of the smallest ring is given, the second ring would somehow relate.
Yes, the question is asking to find the area of the second ring (along with the third, but I figured after figuring out the second, you just mimic the procedure to find the third).
Apparently you didn't understand anything that I said.
The problem states "each successive year's growth creates a ring with the same area as the original circle"
This means that the area of each ring is the same as the original circle, which I have said that many times.
If you want to find the area of the 1st ring, it is given as the same as that of the innermost circle, which is pi sq.inches. The area of the 2nd ring is also pi sq.in.
What I understand from your many questions is you want to find the dimensions of the first ring. And those of the 2nd ring too.
The area of the innermost circle is A1 = pi(1^2) = pi sq.inches
The area of the first ring is pi sq.in. also ....given.
Say the outside radius of the first ring is x inches.
We know that its inner radius is 1 inch
A2 = area of the 1st ring.
So
A2 = (pi)(x^2) -pi(1^2) = pi(x^2 -1^2)
That is equal to pi, so,
pi(x^2 -1) = pi
Divide both sides by pi,
x^2 -1 = 1
x^2 = 2
x = sqrt(2) = 1.4142 inches ------ the outside radius of the first ring.
For the 2nd ring, the same procedure.
Say y = outside radius of the 2nd ring.
We now know that the inner radius of the 1st ring is 1.4142 inches.
So,
A3 = pi(y^2 -(1.4142)^2) = pi
y^2 -(1.4142)^2 = 1
y^2 -2 = 1
y^2 = 3
y = sqrt(3) = 1.732 inches.
Meaning, the 3rd ring will have an outside radius of sqrt(4) = 2 inches.
The 4th will have sqrt(5) inches.
The 5th will have sqrt(6) inches.
.
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The nth ring will have an inner radius of sqrt(n) inches, and an outside radius of sqrt(n+1) inches.