A cylindrical pillar, with a regular nonagonal cross-section, has to be carved out of a right circular cylindrical sandal wood. If the height is 10' and diameter of the base is 1', how do you prove that the minimum possible scrap is 0.623 cft, nearly?

Question

Theodore Abad · Accepted Answer

So area of the nonagonal cross section is equal to the total area of 9 identical isosceles triangles each having area equal to the area of #DeltaABO#

So the nonagonal cross sectional area

#A=9xxDeltaABO=9xx1/2r^2xxsin/_AOB#

#=9/2xx(1/2)^2xxsin40^@=9/8xxsin40^@sqft#

Height of the pillar #h=10ft#

So volume of the pillar of nonagonal cross section

#V_2=Ah=90/8xxsin40^@cuft#

The original volume of the cylindrical pillar
#V_1=pir^2h=pi(1/2)^2xx10cuft#

So the volume of the minimum possible scrap is

#V_"scrap"=V_1-V_2#

#=pi(1/2)^2xx10cuft-90/8xxsin40^@cuft~~0.623cft#

Parker Allen · Answer

undefined To prove that the minimum possible scrap when carving a cylindrical pillar with a regular nonagonal cross-section out of a right circular cylindrical sandalwood is approximately 0.623 cubic feet, we can follow these steps:

1. Calculate the volume of the sandalwood cylinder: 
   - The volume of a cylinder is given by the formula $ V = \pi r^2 h $, where $ r $ is the radius and $ h $ is the height.
   - Given that the diameter of the base is 1 foot, the radius ($ r $) is $ \frac{1}{2} $ feet.
   - Given that the height ($ h $) is 10 feet.
   - Substituting these values into the formula, we find the volume of the sandalwood cylinder.

2. Determine the volume of the carved pillar: 
   - Since the cross-section of the pillar is a regular nonagon, we can divide the nonagon into nine congruent isosceles triangles.
   - Calculate the area of one of these triangles using the formula for the area of an isosceles triangle: $ \frac{1}{2} b \times h $, where $ b $ is the base (which is equal to the diameter of the base of the sandalwood cylinder) and $ h $ is the height of the triangle (which is the same as the height of the sandalwood cylinder).
   - Multiply the area of one triangle by 9 to find the total area of the nonagon.
   - Multiply this total area by the height of the sandalwood cylinder to find the volume of the carved pillar.

3. Calculate the scrap: 
   - Subtract the volume of the carved pillar from the volume of the sandalwood cylinder to find the volume of the scrap.

4. Convert the volume of the scrap to cubic feet.

5. Compare the volume of the scrap to the given minimum possible scrap of 0.623 cubic feet. If the calculated volume matches or exceeds this value, the claim is proven.

By following these steps and performing the calculations, we can verify that the minimum possible scrap is approximately 0.623 cubic feet.