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suppose the radius and height of a cylinder are both doubled. by how much is the surface area increased, by how much is the volume increased

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We start with the surface area equation, S = 2(pi)r2 + 2(pi)rh, if the height and radius were doubled then the knew equation becomes S' = 2(pi)(2r)2 + 2(pi)(2r)(2h) = 4(2(pi)r2 + 4(2(pi)rh) = 4(2(pi)r2 +2(pi)rh).

Examining the final result carefully we notice that the knew surface area S' = 4S so it increases by 4 times its orginal.

V = (pi)r2
      (pi)(2r)2(2h) = 8(pi)r2h = 8*V 

S.A. = 2(pi)r2 + 2(pi)rh 
          2(pi)(2r)2 + 2(pi)(2r)(2h) = 4[2(pi)r2 + 2(pi)rh] = 4 *(S.A.)

In order to find the volume of a cylinder, you would use the formula V=3.14*r2*h.  In the formula, the radius is being squared.  That means that doubling the radius of the cylinder will quadruple the volume.  For example, let's say we have a cylinder with a radius of 2 inches and height of 3 inches.  The volume would be 3.14*22*3 = 3.14*4*3 = 37.68 in2.  However, doubling the radius would have this effect: 3.14*42*3 = 3.14*16*3 = 150.72 in2.  That is four times as large as the original volume.  Now, because the height is not squared in the original formula, doubling the height of a cylinder will just double the volume.  If we put all of that information together (doubling the radius quadruples the volume, while doubling the height doubles the volume), we find that the volume will be 8 times as large.  This is because the radius multiplies the volume by four and the height multiplies the volume by 2, and 4*2=8.