Question

3. Given rectangles and , and given that rectangle is fully rotated about line BC, answer the following questions. (a) Describe the figure generated by the rectangle rotating about line BC. (b) About which line could you rotate rectangle to generate a figure that is similar to the first solid? (c) What are the ratios of the surface areas and volumes of the solids generated from and ? (d) Are and golden rectangles? Why or why not?

Answer 1

It should form a cylinder. 

b) 
thats true. 

c) 
Using Formulas: 
scale factor = 1.5/3 
scale factor = 1/2 

Surface area_small/Surface area_big = (scale factor)^2 
Surface area_small/Surface area_big = (1/2)^2 
Surface area_small/Surface area_big = 1/4 

Volume_small/Volume_big = (scale factor)^3 
Volume_small/Volume_big = (1/2)^3 
Volume_small/Volume_big = 1/8 

Check Understanding: 
lets find the ratio from the small cylinder to the big cylinder. 

S_small/S_big = ( 2π * (r_s)^2 + 2π r_s h_s ) / ( 2π * (r_s)^2 + 2π r_s h_s ) 
S_small/S_big = 2π ( (r_s)^2 + r_s h_s ) / 2π( (r_s)^2 + r_s h_s ) 
S_small/S_big = ( (r_s)^2 + r_s h_s ) / ( (r_s)^2 + r_s h_s ) 
S_small/S_big = ( (1.5)^2 + (1.5) * 2.4 ) / ( (3)^2 + 3 * 4.8 ) 
S_small/S_big = 1/4 <---- true 

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d) 
I will show sort of an explanation: 
http://imageshack.us/photo/my-images/521... 

lets take the longer piece of the big rectangle divide by the shorter. 
AD/AB = 4.8/3 = 1.6 

small rectangle must be since it is similar to. 
PQ/PS = 2.4/1.5 = 1.6 

yes they are since the longer piece to the shorter is close to 1.618