Radius of a Sphere
Date: 01/29/98 at 00:27:35 From: Nino John A. Guerrero Subject: Geometry The radius of the sphere in the exercise is 3 times the radius of the sphere in the examples. What is the ratio of the areas of the two spheres? What is the ratio of their volumes? What possible theorems are suggested by the results?
Date: 01/30/98 at 17:37:04 From: Doctor Bill Subject: Re: Geometry Nino, If the sphere in the examples has a radius of r, then the sphere in the exercise has a radius of 3r, since it is 3 times larger. The formula for the surface area of a sphere is 4*pi*r^2, which will be the area of the sphere in the examples. Since the radius of the sphere in the exercise is 3r, then its surface area is 4*pi*(3r)^2 = 4*pi*9r^2. To find the ratio of the surface areas of the two spheres we divide one formula by the other: (4*pi*9r^2)/(4*pi*r^2). After you simplify by cancelling the common terms in the numerator and the denominator, you should get that the ratio is 9. That means if you increase the radius of a sphere 3 times, you increase the surface area 9 times. Do you see a realationship between the 3 and the 9? What if you increased the radius by 4 times, that is, the new radius is 4r - how much do you think the surface area would increase? (As a hint, notice that in the formula the radius is SQUARED.) Check it out to see by plugging 4r into the formulas above where we had 3r before. The formula for the volume of a sphere is (4/3)*pi*r^3, which is the volume of the sphere in the examples. Once again, the radius of the sphere in the exercise is 3 times larger, so its volume is (4/3)*pi* (3r)^3. Now divide these two formulas as we did above and simplify. You should get that the larger sphere (with radius 3r) has a volume 27 times larger than the smaller sphere. Once again, do you notice a relation between the 3 and the 27? What do you think the ratio of the volumes would be if you increased the radius by 4 times, that, is make the new radius 4r? (Another hint, notice that in the volume formula the radius is raised to the 3rd power.) From the two results above, and the questions I asked, what two theorems do you think you can state now? -Doctor Bill, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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