New streetlights will be installed along a section of the highway. The posts for the streetlights will

be 7.5 m tall and made of aluminum. The city can choose to buy the posts shaped like cylinders or

the posts shaped like rectangular prisms.

The cylindrical posts have a hollow core, with aluminum 2.5 cm thick, and an outer diameter of

53.4 cm. The rectangular-prism posts have a hollow core, with aluminum 2.5 cm thick, and a

square base that measures 40 cm on each side.

The density of aluminum is 2.7 g/cm3, and the cost of aluminum is $0.38 per kilogram.

a) What is the cost of the cylindrical posts, to the nearest cent?

b) What is the cost of the rectangular-prism posts, to the nearest cent?

cylinder: π/4 * (53.4^2 - (53.4-5)^2) * 750 cm^3 * 2.7 g/cm^3 * 1kg/1000g * $0.38/kg = $307.62

prism: (40-5)^2 * 750 cm^3 * 2.7 g/cm^3 * 1kg/1000g * $0.38/kg = $942.64

To calculate the cost of the cylindrical posts, we need to determine the volume of aluminum used and then multiply it by the density of aluminum and the cost per kilogram.

a) Volume of aluminum in the cylindrical post:
The inner radius of the cylindrical post is equal to half the outer diameter minus the thickness of the aluminum layer, which is 2.5 cm.
Inner radius = (53.4 cm / 2) - 2.5 cm = 23.2 cm.

The height of the cylindrical post is given as 7.5 m, which is equal to 750 cm.

Volume of aluminum = π * (outer radius^2 - inner radius^2) * height.
Volume = π * (26.7^2 - 23.2^2) * 750 = 146,226.9 cm^3.

b) Cost of the cylindrical posts:
Density of aluminum = 2.7 g/cm^3.

Mass of aluminum = Volume * Density.
Mass = 146,226.9 cm^3 * 2.7 g/cm^3 = 394,697.13 g.

Cost of aluminum = $0.38 per kilogram = $0.38 / 1000 g.
Cost = 394,697.13 g * $0.38 / 1000 g = $149.91.

Therefore, the cost of the cylindrical posts is $149.91 to the nearest cent.

To calculate the cost of the rectangular-prism posts, we again need to determine the volume of aluminum used and then multiply it by the density of aluminum and the cost per kilogram.

a) Volume of aluminum in the rectangular-prism post:
The height of the rectangular-prism post is given as 7.5 m, which is equal to 750 cm.

Volume of aluminum = base area * height.
Base area = side length^2.
Base area = 40 cm * 40 cm = 1600 cm^2.

Volume = 1600 cm^2 * 750 cm = 1,200,000 cm^3.

b) Cost of the rectangular-prism posts:
Density of aluminum = 2.7 g/cm^3.

Mass of aluminum = Volume * Density.
Mass = 1,200,000 cm^3 * 2.7 g/cm^3 = 3,240,000 g.

Cost of aluminum = $0.38 per kilogram = $0.38 / 1000 g.
Cost = 3,240,000 g * $0.38 / 1000 g = $1231.20.

Therefore, the cost of the rectangular-prism posts is $1231.20 to the nearest cent.

To find the cost of the cylindrical posts and the rectangular-prism posts, we need to calculate the volume of aluminum used in each type of post, and then multiply it by the cost per kilogram of aluminum.

First, let's calculate the volume of aluminum used in each type of post.

For the cylindrical post:
1. Calculate the inner radius of the hollow core:
Inner diameter = Outer diameter - 2 * thickness
Inner radius = Inner diameter / 2

2. Calculate the volume of the hollow core:
Volume of the hollow core = π * (Inner radius)^2 * height

3. Calculate the volume of the post without the hollow core:
Volume of the post without hollow core = π * (Outer radius)^2 * height

4. Calculate the volume of aluminum used in the cylindrical post:
Volume of aluminum used = Volume of the post without hollow core - Volume of the hollow core

For the rectangular-prism post:
1. Calculate the volume of the hollow core:
Volume of the hollow core = (Base side)^2 * height

2. Calculate the volume of the post without the hollow core:
Volume of the post without hollow core = (Base side + thickness)^2 * height

3. Calculate the volume of aluminum used in the rectangular-prism post:
Volume of aluminum used = Volume of the post without hollow core - Volume of the hollow core

Now, let's plug in the given values and calculate the volume of aluminum used in each type of post.

For the cylindrical post:
Outer diameter = 53.4 cm
Thickness = 2.5 cm
Height = 7.5 m

1. Calculate the inner radius of the hollow core:
Inner diameter = 53.4 cm - 2 * 2.5 cm = 48.4 cm
Inner radius = 48.4 cm / 2 = 24.2 cm

2. Calculate the volume of the hollow core:
Volume of the hollow core = π * (24.2 cm)^2 * 750 cm = 142,897 cm^3

3. Calculate the volume of the post without the hollow core:
Outer radius = 53.4 cm / 2 = 26.7 cm
Volume of the post without hollow core = π * (26.7 cm)^2 * 750 cm = 449,244 cm^3

4. Calculate the volume of aluminum used in the cylindrical post:
Volume of aluminum used = 449,244 cm^3 - 142,897 cm^3 = 306,347 cm^3

Now, let's calculate the volume of aluminum used in the rectangular-prism post.

For the rectangular-prism post:
Base side = 40 cm
Thickness = 2.5 cm
Height = 7.5 m

1. Calculate the volume of the hollow core:
Volume of the hollow core = (40 cm)^2 * 750 cm = 1,200,000 cm^3

2. Calculate the volume of the post without the hollow core:
Volume of the post without hollow core = (40 cm + 2.5 cm)^2 * 750 cm = 1,287,500 cm^3

3. Calculate the volume of aluminum used in the rectangular-prism post:
Volume of aluminum used = 1,287,500 cm^3 - 1,200,000 cm^3 = 87,500 cm^3

Next, we need to convert the volume of aluminum used into kilograms.

1 cm^3 = 1 g
87,500 cm^3 = 87,500 g = 87.5 kg

Finally, we can calculate the cost of each type of post by multiplying the volume of aluminum used by the cost per kilogram.

Cost of the cylindrical posts = 306,347 cm^3 * (2.7 g/cm^3 / 1000) * $0.38 ≈ $310.69
Cost of the rectangular-prism posts = 87.5 kg * $0.38 ≈ $33.25

Therefore,
a) The cost of the cylindrical posts is approximately $310.69.
b) The cost of the rectangular-prism posts is approximately $33.25.