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Session 8: Solutions
 
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Solutions for Session 8 Homework

See solutions for Problems: H1 | H2 | H3 | H4 | H5 | H6


Problem H1

a. 

As long as the base is a regular polygon, its shape begins to approximate a circle. The triangular sides become increasingly small. In time, the pyramid becomes indistinguishable from a cone.

b. 

The volume formula is still V = (1/3)Bh, but since the base becomes a circle, the volume becomes V = (1/3)r2h.

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Problem H2

If the diameter of the ball is 1 ft., then the radius is 0.5 ft. Also, the height of the box is 1 ft. So the volume of the sphere is (4/3) (0.5)3 = 0.52 ft3 (rounded to hundredths using the key on your calculator). The volume of the box is 1 ft3. To obtain the volume of box that is foam, subtract the volume of the sphere from the volume of the box (1 - 0.52 = 0.48 ft3). So 0.48 ft3 is filled with foam. You could convert everything to inches to solve this as well.

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Problem H3

a. 

The shorter, wider cylinder has the greatest volume. In the formula for the volume of a cylinder, V = r2h, notice that the radius is squared and the height is not. Using the larger dimension of the paper as a circumference of the base produces a larger radius, and in turn, an exponentially larger volume, and vice versa: Using a smaller dimension of the paper as a circumference of the base produces a smaller radius, and in turn, an exponentially smaller volume.

b. 

Again, the shorter, wider cylinder has the greatest surface area. All will have equal lateral surface area (not including the top and bottom), since the same paper is being used. For total surface area, add the area of the bases -- so the problem boils down to which has the largest base area. As described in the solution to part (a), the shorter, wider cylinder has the largest base area.

c. 

The cylindrical container has the greater volume, as long as they both have the same lateral surface. If only the heights are known, then there is no comparison to make -- one could have a much larger base area than the other.

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Problem H4

Box A is 1 by 1 by 6. Its volume is 6 cubic units.
Box B is 1 by 3 by 3. Its volume is 9 cubic units.
Box C is 2 by 2 by 4. Its volume is 16 cubic units.

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Problem H5

a. 

Answers will vary. Your answers should reveal a royal cubit to be about 52.4 cm or 20.62 in.

b. 

Since a cubit is just over 0.5 m, we would expect there to be just under 2 cubits in each dimension in 1 m3. This means the volume is just under 8 cubic cubits. Based on the value of 52.4 cm in 1 cubit, there are 6.95 cubic cubits in a cubic meter. There are 1,000,000 cm3 in a cubic meter.

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Problem H6

Answers will vary. A fathom is roughly 2 m, so a cubic fathom is about 8 m3. If one piece of firewood measures 30 cm by 10 cm by 10 cm, one piece of firewood is 3,000 cm3. Since there are 1,000,000 cm3 in a cubic meter, then about 333 pieces of firewood fit in 1 m3, and about eight times that (roughly 2,600 pieces) fit in a cubic fathom.

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