Question 305295: I don't understand how ratios work with solid figures. If I am given the radius of one cylinder is 12cm and another cylinder's radius is 4cm how do I find the ratio of the circumference or the ratio of surface areas?
If I'm given two prisms with the scale factor 1:2 and the little prism's surface area equals 208ins then how do I find the surface area of the larger solid?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! You picked a very hard example with Surface Areas of a cylinder because the ratio of the surface area of a cylinder is not consistently the same across all cylinders regardless of their dimensions.
Let's try something simpler, like the volume of a cylinder.
You first need to find the formula of the volume of a cylinder in terms of the radius of the cylinder.
You either know it or you look it up in your text or on the web or ask somebody who knows.
For a cylinder, the formula for the volume of a cylinder would be equal to 2 * pi * r^2 * h where r is the radius and h is the height of the cylinder.
Now that you know the formula for the volume of a cylinder, you work the problem as follows:
The radius of the second cylinder is equal to 3 times the radius of the first cylinder.
If the formula for the volume of the first cylinder is equal to:
pi * r^2 * h
Then the volume of the second cylinder must be equal to:
pi * (3*r)^2 * h which is equivalent to:
pi * 9 * r^2 * h which is equivalent to:
9 * pi * r^2 * h
The ratio of the volume of the first cylinder to the second cylinder becomes:
(pi * r^2 * h) / (9 * pi * r^2 * h)
The (pi * r^2 * h) in the numerator and the denominator cancel out and you are left with:
1 / 9
This means that, if the ratio of the radius of the first cylinder to the radius of the second cylinder is equal to 1 / 3, then the ratio of the volume of the first cylinder to the second cylinder is equal to 1 / 9.
Since all variables canceled out, then this ratio will hold true for all sizes of cylinders where the radius of the second cylinder is equal to 3 times the radius of the first cylinder and the height of the second cylinder is the same as the height of the first cylinder.
Assume the radius of the first cylinder is equal to 7 and the height of the first cylinder is equal to 15.
The volume of the first cylinder is equal to pi * 7^2 * 15 = 735 * pi.
The radius of the second cylinder will be 3 * 7 = 21 and the height of the second cylinder will still be 15.
The volume of the second cylinder is equal to pi * 21^2 * 15 = 6615 * pi.
6615 * pi is equal to 9 times 735 * pi as it should be if the ratio formulas hold.
Since 6615 / 735 = 9, then the ratio holds.
The statement that says the volume of the second cylinder will be 9 times the volume of the first cylinder if the radius of the second cylinder is 3 times the radius of the first cylinder is confirmed to be true for all measurements of a cylinder as long as you stay within the assumption that the height will remain the same.
Your problem involving the surface area of a cylinder is much more difficult to solve because the ratio of the surface area of the first cylinder to the surface area of the second cylinder will be dependent on the actual radius and the actual height and will be different if you change those dimensions.
Here's why.
the formula of the surface area of a cylinder is equal to:
(2 * pi * r^2) + (2 * pi * r * h).
This can be factored to get:
2 * pi * r * (r + h)
If you allow this formula for the first cylinder, and the radius of the second cylinder is equal to 3 times the radius of the first cylinder, then the formula for the surface area of the second cylinder will be equal to:
(2 * pi * (3*r)^2) + (2 * pi * (3*r) * h)
This becomes:
(2 * pi * 9 * r^2) + (2 * pi * 3 * r * h)
This can be simplified to:
(18 * pi * r^2) + (6 * pi * r * h)
This can be factored to get:
6 * pi * r * ((3 * r) + h)
The ratio of the surface area of the first cylinder to the surface area of the second cylinder becomes:
(2 * pi * r * (r + h)) / (6 * pi * r * ((3 * r) + h))
The pi * r in the numerator and denominator cancel out to get:
(2 * (r + h)) / (6 * (3*r + h)) which can be further reduced to:
(r + h) / (3 * (3*r + h)) which can be further reduced to:
(r + h) / (9*r + 3*h)
The r and the h cannot be canceled out in this equation which means that this ratio depends on the specific value of r and h.
If you change either one, you change the ratio.
the ratio is not valid for all measurements of cylinders.
It is valid for specific measurements only.
For example, assume the radius = 7 and the height = 5.
The ratio of (r + h) / (9r + 3h) becomes (7+5) / (9*7 + 3*5) which becomes:
12 / 78.
What this says is that, if you have a cylinder with a radius of 7 and a height of 5, and you triple the radius, then the ratio of the surface area of the first cylinder to the surface area of the second cylinder will be equal to 12 / 78.
If, however, you have a cylinder with a radius of 7 and a height of 9, then the ratio of the surface area of the first cylinder to the surface area of the second cylinder will be different.
In that case if will be equal to:
(7 + 9) / (9*7 + 3*9) which would be equal to:
16 / 90
16 / 90 is not the same ratio as 12 / 78.
Change the height and the ratio changes.
Likewise, if you have a cylinder with a radius of 5 and a height of 5, then the ratio of the surface area of the first cylinder to the surface area of the second cylinder will be equal to:
(5 + 5) / (9*5 + 3*5) which would be equal to:
10 / 60 which is not the same ratio as 12 / 78
Change either the height or the radius and the ratio of the surface area of the first cylinder to the surface area of the second cylinder will not always be the same.
You can find it if you are given the specific measurements, but not in the general case that applies to all measurements.
With the volume of the cylinder, you were able to find a ratio that applied to all measurements.
Your last question was as follows:
If I'm given two prisms with the scale factor 1:2 and the little prism's surface area equals 208 inhes then how do I find the surface area of the larger solid?
The answer to this depends on what the prism is.
Is it a rectangular prism, a cube?, hexagonal prism?
The scaling factor applies to the dimensions of the prism.
Let's assume it's a rectangular prism.
The volume of a rectangular prism is equal to L * W * H where L is the length and W is the width and H is the height.
The surface area of a rectangular prism is equal to 2*L + 2*W + 2*H.
This is equivalent to 2 * (L + W + H)
You are given that the scaling factor is 1 to 2.
This means that, if the volume of the first prism is equal to L * W * H, then the volume of the second prism is equal to (2*L) * (2*W) * (2*H) which would be equal to 8 * (L * W * H)
If the scaling factor is 1 to 2, then the volume of the larger prism will be 8 times the volume of the smaller prism.
If the scaling factor is 1 to 2, and the surface area of the smaller prism is equal to 2*L + 2*W + 2*H then the surface area of the larger prism will be equal to 2*2*L + 2*2*W + 2*2*H.
2*L + 2*W + 2*H is equivalent to 2 * (L + W + H).
2*2*L + 2*2*W + 2*2*H is equivalent to 4*L + 4*W + 4*H which is equivalent to 4 * (L + W + H)
The ratio of the surface area of the larger prism to the surface area of the smaller prism will be equal to:
(4 * (L + W + H)) / (2 * (L + W + H))
The (L + W + H) in the numerator and the denominator cancel out and the ratio becomes 4 / 2 which becomes 2 to 1.
This means that the surface area of the larger prism will be 2 times the surface area of the smaller prism.
This means that, if the surface area of the smaller prism is equal to 208 inches, then the surface area of the larger prism will be 416 inches.
Let's assume the dimensions of the smaller prism are:
L = 52
W = 13
H = 39
The Surface area of the prism would be equal to 2*52 + 2*13 + 2*39 = 208
If the scaling factor is 1 to 2, then each dimension will double to get:
L = 104
W = 26
H = 78
The surface area of the larger prism will be equal to 2*104 + 2*26 + 2*78 = 416
416 is equal to 2 times 208 so the ratio is correct.
In all of these problems, the minimum amount of information you needed was:
1. What is the shape of the prism being represented?
2. What is the formula for the prism that is needed to solve the problem?
With the prism we just went through, we assumed the prism was a rectangular prism and then we found the formula for the surface area of the prism.
With the cylinder, we just went through, we knew the shape was a cylinder and we determined that we needed the formula for the volume of a cylinder for the first part of the example, and that we needed the formula for the surface area of a cylinder for the second part of the example.
If you are given a specific problem to solve and you are stuck, send me an email and I'll take a look at if for you.
My email address is dtheophilis@yahoo.com
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