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A square prism with a vertical rectangular portion removed from the bottom middle. The side of the square prism is 10 feet. The height of the portion removed from the middle, from the base upward, is 5 feet. The width of the remaining two legs on the left and right of the removed portion is 4 feet each.

A giant model of a magnet is being built for a museum exhibit. What is the total surface area of the prop that will need to be painted if the depth of the magnet is 3 feet?

(1 point)
Responses

180 ft.2
180 ft. squared

410 ft.2
410 ft. squared

370 ft.2
370 ft. squared

330 ft.2

1 answer

To find the total surface area of the square prism with a rectangular portion removed, first, we need to visualize and calculate various components of the figure.

  1. Dimensions of the square prism:

    • Base: 10 feet x 10 feet
    • Height: The height is not specified explicitly for the total prism, but we assume it to be the same height of the removed portion, which is 5 feet for the removed section. The overall height of the prism can be inferred to be 5 feet since that's the height mentioned for the removed section and typically such prisms match this height.

  2. Dimensions of the rectangular portion removed:

    • Removed Section: Width = 2 feet (since 10 feet total - 2*4 feet for the remaining sides), Height = 5 feet, Depth = 3 feet.
    • The removed portion effectively creates a rectangular hole through the prism.

  3. Surface Area Calculations:

    • Total surface area (without the removed section) for a full prism is comprised of:
      • 2 base areas + 4 lateral (side) areas.
    • Each base area = 10 ft x 10 ft = 100 ft²; there are 2 bases = 100 ft² * 2 = 200 ft².
    • Lateral area: The height is 5 feet, and there are four rectangles to consider; however, the back of the removed section needs to be considered separately.
      • The left and right lateral sides: 2 sides (10 ft height x 4 ft width) = 2 * (5 ft * 4 ft) = 40 ft².
      • The top & bottom sides (remaining) = 5 ft x 10 ft for the remaining sides and 2 sides of the back of the removable section = 5 ft x 2 ft for where the rectangle is removed.
        • Total = (5 ft * 10 ft * 2) for the length sides + (2*5 ft * 2ft) back = 100 ft² + 20 ft² = 120 ft².
    • Subtract the area that is now missing from the surface:
      • The removed section allows us to add the top of the hole.

    So total area without the hole minus the back portions:

    • With all components calculated:
      • Total Surface Area = \(200 + 120 + 40 - 10 = 330 ft²\).

Based on the given responses, the total surface area of the prop that will need to be painted is: 330 ft².