Problem 7 (a)
Pick any two values you want for the radius and height (r and h). Repeats are allowed.
I'll go with r = 1 and h = 1
Compute the surface area of the cylinder with these parameters.
SA = 2pi*r^2 + 2*pi*r*h
SA = 2pi*1^2 + 2pi*1*1
SA = 2pi + 2pi
SA = 4pi
Now double the radius. We go from r = 1 to r = 2. Keep the height the same and compute the new surface area.
SA = 2pi*r^2 + 2*pi*r*h
SA = 2pi*2^2 + 2*pi*2*1
SA = 8pi + 4pi
SA = 12pi
The jump from 4pi to 12pi is "times 3".
Answer: The surface area triples
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Problem 7 (b)
Let's go back to r = 1 and h = 1 again.
LSA = lateral surface area of the cylinder
LSA = 2*pi*r*h
LSA = 2*pi*1*1
LSA = 2pi
Now double r, but keep h the same
LSA = 2*pi*r*h
LSA = 2*pi*2*1
LSA = 4pi
Going from 2pi to 4pi is a jump of "times 2"
Answer: The lateral area doubles
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Problem 7 (c)
We'll go back to r = 1 and h = 1.
Compute the volume of this cylinder.
V = pi*r^2*h
V = pi*1^2*1
V = pi
Then compute the volume with double the radius, but the same height
V = pi*r^2*h
V = pi*2^2*1
V = 4pi
The volume has quadrupled since we have gone from pi to 4pi.
Answer: Volume quadruples
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Problem 8
The volume of the block, before the half cylinders are removed, is 10*3*8 = 240 cubic inches.
The two half cylinders removed can be rearranged to combine to a full cylinder. This empty space cylinder has a radius of 8/2 = 4 inches and height 3 inches. This means we'll be kicking out the volume of pi*r^2*h = pi*4^2*3 = 48pi cubic inches of empty space.
The resulting volume of this weird looking solid is 240-48pi cubic inches.
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The front panel is a 3 in by 10 in rectangle. It has area of 3*10 = 30 square inches. The same goes for the back wall.
The top or ceiling panel is going to involve us first finding the area of the 8 by 10 rectangle (if you didn't remove those circular portions). So we have 8*10 = 80 square inches to start with. Then we kick out the circle of radius 4. So the top and bottom panels have area of 80 - 16pi square inches each.
The left and right curved walls are the lateral sides of a full cylinder of radius 4 inches and height 3 inches. This is when we combine the two curved sides to form one cylinder.
LSA = 2*pi*r*h = 2*pi*4*3 = 24pi
This means each curved wall is 24pi/2 = 12pi square inches in surface area
Now add up the 6 sides mentioned
(front)+(back)+(top)+(bottom)+(left)+(right)
(30)+(30)+(80 - 16pi)+(80 - 16pi)+(12pi)+(12pi)
220-8pi
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Answers:
Surface area = 220-8pi square inches
Volume = 240-48pi cubic inches.