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Tyler is preparing to paint the shed in his backyard. The shed is a rectangular prism. He plans to paint all sides EXCEPT the side that sits on the ground. The length of the shed is 16 feet. The width of the shed is 25.5 feet. The height of the shed it 8 feet. In order to know how much paint to buy, Tyler must find the surface area that needs to be painted. Explain the steps to find the surface area that needs to be painted AND tell what the surface area is that needs to be painted. (4 points)

1 answer

To find the surface area that needs to be painted for Tyler's shed, we will calculate the total surface area of the rectangular prism and then subtract the area of the bottom face (the side that sits on the ground).

Steps to Calculate the Surface Area to be Painted:

  1. Identify the dimensions of the shed:

    • Length (l) = 16 feet
    • Width (w) = 25.5 feet
    • Height (h) = 8 feet

  2. Calculate the total surface area of the rectangular prism: The total surface area (SA) of a rectangular prism is calculated using the formula: \[ SA = 2lw + 2lh + 2wh \] where:

    • \(lw\) is the area of the top and bottom faces,
    • \(lh\) is the area of the front and back faces, and
    • \(wh\) is the area of the left and right faces.

    Substituting in the dimensions: \[ SA = 2(16 \times 25.5) + 2(16 \times 8) + 2(25.5 \times 8) \]

  3. Calculate each part of the formula:

    • \(lw = 16 \times 25.5 = 408\) square feet (top and bottom faces)
    • \(lh = 16 \times 8 = 128\) square feet (front and back faces)
    • \(wh = 25.5 \times 8 = 204\) square feet (left and right faces)

    Putting these into the surface area formula: \[ SA = 2(408) + 2(128) + 2(204) \] \[ SA = 816 + 256 + 408 \] \[ SA = 1480 \text{ square feet} \]

  4. Subtract the area of the bottom face: The area of the bottom face (which Tyler does not plan to paint) is the same as the top face (since they are identical): \[ \text{Area of bottom face} = lw = 16 \times 25.5 = 408 \text{ square feet} \]

  5. Calculate the surface area that needs to be painted: \[ \text{Surface Area to be Painted} = SA - \text{Area of Bottom Face} \] \[ \text{Surface Area to be Painted} = 1480 - 408 = 1072 \text{ square feet} \]

Final Answer:

The surface area that Tyler needs to paint is 1072 square feet.