You are able to purchase one of the two different types of hay bales: cylinder shaped or rectangular shaped. The cylinder hay bale is 4 feet high and has a diameter of 6 feet. The rectangular hay bale has a width of 2 feet, height of 1 ½ feet, and length of 3 feet. Note: You must purchase whole bales. 1. Approximately how many rectangular shaped bales would you need to purchase to have the same amount of hay as one cylinder shaped bale? 2. You need 16,000 ft3 of hay to last through the year. Determine how many of each hay bale you would need to purchase if you purchase either all rectangular bales or all cylinder shaped bales. If one cylinder shaped bale costs $20 and one rectangular bale costs $2.75, which will be the better value? 3. Your barn measures 36 feet long, 36 feet wide, and 12 feet high. What is the greatest number of cylinder shaped bales that will fit in the barn? rectangular bales? Which type of hay bale will allow you to fit a greater volume of hay in the barn?
*PLease Show Work*​

Answers

Answer 1

Answer:

  1. 13 rectangular bales

  2. 142 cylindrical bales (better value); 1778 rectangular bales

  3. 108 cylindrical bales; 1728 rectangular bales (more volume)

Explanation:

1. The volume of a cylindrical bale is given by the formula ...

  V = πr²h

  V = π(3 ft)²(4 ft) = 36π ft³ ≈ 113.1 ft³

The volume of a rectangular bale is ...

  V = LWH

  V = (3 ft)(2 ft)(1.5 ft) = 9 ft³

The number of rectangular bales required to match the volume of a cylindrical bale is ...

  (113.1 ft³)/(9 ft³) ≈ 12.57

You would need to purchase 13 rectangular bales to have the same amount of hay as one cylindrical bale.

__

2. For 16,000 ft³ of hay, you need ...

  16000/113.1 = 141.47 ≈ 142 cylindrical bales

  16000/9 = 1777.8 ≈ 1778 rectangular bales

The relative cost of a cylindrical bale is ...

  20/2.75 = 7.27 . . . times the rectangular bale cost

times the cost of a rectangular bale. Since it has 12.57 times the volume, the cylindrical bale is the better value.

__

3. If the cylindrical bales are placed on a 6-foot grid and stacked 3 high, then 6×6×3 = 108 of them can be put in the barn.*

The barn volume is (36 ft)(36 ft)(12 ft) = 15,552 ft³ = 1728 times the 9 ft³ volume of a rectangular bale.

Since there is no wasted space for a rectangular bale in the rectangular space, a greater volume of hay will fit with rectangular bales.

  108 cylindrical bales will fit   (76% of annual need)

  1728 rectangular bales will fit   (97% of annual need)

_____

* The space utilization on a square grid is π/4 ≈ 78.5%. About the best that can be achieved is 90.7% with hexagonal packing. In a rectangular barn, that density is not achievable. Something between those values may be achievable if the cylindrical bales are deformed a little bit.

Answer 2

Answer:

1. 13 rectangular bales

2. 142 cylindrical bales

3. 108 cylindrical bales

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__

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