Question 1178894: compressair company provides cylindrical tanks of compressed air for divers. There standard tank is a cylinder with a volume of 0.015 m^3. Each cylinder is 75cm long and is designed to fit into a special diver's pack.
a) what is the diameter of the cylinder, to the nearest centimetre
b) compressair staff are designing a new cylinder that also has a volume of 0.015 m^3, but uses less material to make. What are the dimensions of the new cylinder, to the nearest tenth of a centimetre? What is the surface area, to the nearest square centimetre?
c)compressair plans to ship 20 of its standard 75-cm-long cylinders upright in a closed rectangular reinforced cardboard box. Staff are considering packing the cylinders in identical rows or in staggered rows, as shown below. Which packing arrangement will require the box of least volume and surface area
box a: identical 5 on the top 4 going down
box b: staggered 5 on the top 4 going down
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step:
**a) Diameter of the Standard Cylinder:**
1. **Volume Formula:** The volume of a cylinder is V = πr²h, where V is volume, r is radius, and h is height (length).
2. **Given Values:**
* V = 0.015 m³
* h = 75 cm = 0.75 m (convert to meters for consistent units)
3. **Solve for Radius (r):**
* 0.015 = πr²(0.75)
* r² = 0.015 / (0.75π)
* r² ≈ 0.0063661977
* r ≈ √0.0063661977 ≈ 0.079788 m
4. **Convert Radius to Centimeters:**
* r ≈ 0.079788 m * 100 cm/m ≈ 7.9788 cm
5. **Calculate Diameter (d):**
* d = 2r ≈ 2 * 7.9788 cm ≈ 15.9576 cm
6. **Round to Nearest Centimeter:**
* d ≈ 16 cm
**Therefore, the diameter of the standard cylinder is approximately 16 cm.**
**b) Dimensions of the New Cylinder (Less Material):**
To minimize material, we want to minimize the surface area for a given volume. This occurs when the height of the cylinder is equal to the diameter (h = 2r).
1. **Volume Formula:** V = πr²h
2. **Given Values:**
* V = 0.015 m³
* h = 2r
3. **Substitute h = 2r:**
* 0.015 = πr²(2r)
* 0.015 = 2πr³
* r³ = 0.015 / (2π)
* r³ ≈ 0.002387324
* r ≈ ³√0.002387324 ≈ 0.13365 m
4. **Convert Radius to Centimeters:**
* r ≈ 0.13365 m * 100 cm/m ≈ 13.365 cm
5. **Calculate Height (h):**
* h = 2r ≈ 2 * 13.365 cm ≈ 26.73 cm
6. **Round to Nearest Tenth of a Centimeter:**
* r ≈ 13.4 cm
* h ≈ 26.7 cm
7. **Calculate Surface Area (SA):**
* SA = 2πr² + 2πrh
* SA = 2π(13.365)² + 2π(13.365)(26.73)
* SA ≈ 2π(178.623225) + 2π(357.24645)
* SA ≈ 1122.38 + 2244.75
* SA ≈ 3367.13 cm²
8. **Round to Nearest Square Centimeter:**
* SA ≈ 3367 cm²
**Therefore, the new cylinder dimensions are approximately 13.4 cm radius and 26.7 cm height. The surface area is approximately 3367 cm².**
**c) Packing Arrangement (Least Volume and Surface Area):**
1. **Standard Cylinder Dimensions:**
* Diameter (d) = 16 cm
* Radius (r) = 8 cm
* Height (h) = 75 cm
2. **Box A (Identical Rows):**
* Dimensions:
* Length: 5 * d = 5 * 16 cm = 80 cm
* Width: 4 * d = 4 * 16 cm = 64 cm
* Height: 75 cm
* Volume: 80 cm * 64 cm * 75 cm = 384,000 cm³
3. **Box B (Staggered Rows):**
* Dimensions:
* Length: 5 * d = 80 cm
* Width: d + 3 * (d * √3 / 2) = 16 + 3 * (16 * √3 / 2) ≈ 16 + 41.57 = 57.57 cm
* Height: 75 cm
* Volume: 80 cm * 57.57 cm * 75 cm = 345,420 cm³
**Comparison:**
* **Volume:** Box B (staggered) has a smaller volume (345,420 cm³) compared to Box A (384,000 cm³).
* **Surface Area:** Box B will also have a smaller surface area, as it requires a smaller box overall.
**Conclusion:**
The **staggered packing arrangement (Box B)** will require the box of least volume and surface area.
Answer by ikleyn(52778)  (Show Source):
You can put this solution on YOUR website! .
compressair company provides cylindrical tanks of compressed air for divers.
There standard tank is a cylinder with a volume of 0.015 m^3.
Each cylinder is 75cm long and is designed to fit into a special diver's pack.
a) what is the diameter of the cylinder, to the nearest centimetre
b) compressair staff are designing a new cylinder that also has a volume of 0.015 m^3,
but uses less material to make. What are the dimensions of the new cylinder,
to the nearest tenth of a centimetre? What is the surface area, to the nearest square centimetre?
c)compressair plans to ship 20 of its standard 75-cm-long cylinders upright
in a closed rectangular reinforced cardboard box. Staff are considering packing
the cylinders in identical rows or in staggered rows, as shown below.
Which packing arrangement will require the box of least volume and surface area
box a: identical 5 on the top 4 going down
box b: staggered 5 on the top 4 going down
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
First, nothing is shown below.
Second, part (c) of this problem looks like a JOKE or a TRAP.
Why did I say it ?
Because the scheme 5 + 4 + 5 + 4 allows packing only 5 + 4 + 5 + 4 = 18 cylinders,
and 2 cylinders of 20 remain unpacked.
So, EITHER the problem formulation is skew and curved OR the solution in the post by @CPhill is incorrect.
Thank you for making me smiling !
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