A firm has commissioned an IE to design a trash can with a capacity of 30 gallons. The can is in the form of a right circular cylinder. The bottom is stamped from a circular disk whose radius is 1 in greater than that of the can. The extra inch is turned down to form a rim so that the bottom does not touch the ground. The side fits down over this rim. The top is stamped similar from a disk whose radius is 2 inch greater than that of the can. A heavier material should be used for the top and bottom. The costs will be 4/3 as much per square inch as the material for the sides. Find the relative proportions of the can per minimum cost.
A firm has commissioned an IE to design a trash can with a capacity of 30 gallons. The can is in the form of a right circular cylinder. The bottom is stamped from a circular disk whose radius is 1 in greater than that of the can. The extra inch is turned down to form a rim so that the bottom does not touch the ground. The side fits down over this rim. The top is stamped similar from a disk whose radius is 2 inch greater than that of the can. A heavier material should be used for the top and bottom. The costs will be 4/3 as much per square inch as the material for the sides. Find the relative proportions of the can per minimum cost.
Chapter5: Exponential And Logarithmic Functions
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![**Designing a Cost-Efficient Trash Can**
A firm has tasked an Industrial Engineer (IE) with creating a trash can that can hold 30 gallons. This trash can will take the shape of a right circular cylinder. Below are the specifications and constraints for the design and material costs:
1. **Bottom Design**:
- The base of the trash can is to be stamped from a circular disk.
- The radius of this disk is 1 inch greater than the radius of the can itself.
- The extra inch will be turned down to form a rim, ensuring the bottom does not touch the ground.
2. **Side Design**:
- The side of the trash can will fit over the aforementioned rim.
3. **Top Design**:
- The top of the trash can will be made in a similar fashion to the bottom.
- The disk for the top will have a radius 2 inches greater than the radius of the can.
4. **Material Specifications**:
- Heavier material should be used for the top and bottom of the can.
7. - The cost of the material for the top and bottom will be 4/3 times the cost per square inch of the material used for the sides.
**Objective**:
- Determine the optimal relative proportions for the dimensions of the trash can to minimize the production cost.
**Volume and Cost Calculations**:
In order to design a cost-effective trash can, the IE will need to use volume equations for a cylinder and cost calculations based on the material specifications. The total cost will need to be minimized while keeping the volume fixed at 30 gallons.
**Detailed Steps**:
1. Express the volume in terms of the radius (r) and height (h) of the cylinder:
\[ V = \pi r^2 h \]
Where V = 30 gallons.
2. Convert gallons to cubic inches (since 1 gallon ≈ 231 cubic inches):
\[ 30 \text{ gallons} = 30 \times 231 \text{ cubic inches} \]
3. Calculate the surface area for the bottom, top, and sides using the given radius adjustments.
4. Incorporate the cost ratios for the materials of different parts.
By balancing these equations and solving for the minimum cost, the optimal proportions of the trash can can be determined efficiently. This analysis ensures that while the trash can is functional and meets the volume requirement, it also](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1beaffe7-9d7a-42b9-9107-dccba3680f89%2Ff7fa992c-d747-443b-8fbf-c0deaf1e359a%2Fuvnnb0d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Designing a Cost-Efficient Trash Can**
A firm has tasked an Industrial Engineer (IE) with creating a trash can that can hold 30 gallons. This trash can will take the shape of a right circular cylinder. Below are the specifications and constraints for the design and material costs:
1. **Bottom Design**:
- The base of the trash can is to be stamped from a circular disk.
- The radius of this disk is 1 inch greater than the radius of the can itself.
- The extra inch will be turned down to form a rim, ensuring the bottom does not touch the ground.
2. **Side Design**:
- The side of the trash can will fit over the aforementioned rim.
3. **Top Design**:
- The top of the trash can will be made in a similar fashion to the bottom.
- The disk for the top will have a radius 2 inches greater than the radius of the can.
4. **Material Specifications**:
- Heavier material should be used for the top and bottom of the can.
7. - The cost of the material for the top and bottom will be 4/3 times the cost per square inch of the material used for the sides.
**Objective**:
- Determine the optimal relative proportions for the dimensions of the trash can to minimize the production cost.
**Volume and Cost Calculations**:
In order to design a cost-effective trash can, the IE will need to use volume equations for a cylinder and cost calculations based on the material specifications. The total cost will need to be minimized while keeping the volume fixed at 30 gallons.
**Detailed Steps**:
1. Express the volume in terms of the radius (r) and height (h) of the cylinder:
\[ V = \pi r^2 h \]
Where V = 30 gallons.
2. Convert gallons to cubic inches (since 1 gallon ≈ 231 cubic inches):
\[ 30 \text{ gallons} = 30 \times 231 \text{ cubic inches} \]
3. Calculate the surface area for the bottom, top, and sides using the given radius adjustments.
4. Incorporate the cost ratios for the materials of different parts.
By balancing these equations and solving for the minimum cost, the optimal proportions of the trash can can be determined efficiently. This analysis ensures that while the trash can is functional and meets the volume requirement, it also
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