Ask a New Question
Menu

Use the image to answer the question.

This image is a diagram showing a 3D L-shaped prism. The dimensions are given in inches. The prism is composed of two rectangular prisms joined at a right angle. The longer section measures 16 inches in length, 6 inches in height, and 4 inches in width. The shorter section is perpendicular to the longer section and measures 10 inches in height and 4 inches in width and depth. The diagram assists in visualizing this three-dimensional shape and calculating its volume or surface area, a common task in geometry problems.

Nikolai is making an L-shaped platform for his collection of action figures. What is the surface area of the platform if it is 3 in. thick? (1 point) in.2

1 answer

To calculate the surface area of the L-shaped platform that is 3 inches thick, we first need to determine the surface area of the two sections of the L-shaped prism and then account for the thickness.

Dimensions of the L-shaped prism:

  1. Longer section:
    • Length = 16 inches
    • Height = 6 inches
    • Width = 4 inches
  2. Shorter section:
    • Height = 10 inches
    • Width = 4 inches
    • Depth = 4 inches (since it's a square base)

Volume and Surface Area Calculations:

  1. Surface Area of the Longer Section:

    • The surface area of a rectangular prism is calculated using the formula: \[ SA = 2(lw + lh + wh) \]
    • For the longer section: \[ SA_{\text{long}} = 2(16 \times 4 + 16 \times 6 + 4 \times 6) \] \[ = 2(64 + 96 + 24) = 2(184) = 368 \text{ in}^2 \]

  2. Surface Area of the Shorter Section:

    • Using the same surface area formula:
    • For the shorter section:
      • Length = 4 inches (width)
      • Height = 10 inches
      • Width (depth) = 4 inches \[ SA_{\text{short}} = 2(4 \times 4 + 4 \times 10 + 10 \times 4) \] \[ = 2(16 + 40 + 40) = 2(96) = 192 \text{ in}^2 \]

  3. Total Surface Area of Both Sections (not considering the overlap): \[ SA_{\text{total}} = SA_{\text{long}} + SA_{\text{short}} = 368 + 192 = 560 \text{ in}^2 \]

  4. Adjusting for Thickness (3 inches thick):

    • The thickness does not extend the surface area of the bottom since it's a platform, but we need to consider the four additional sides created by the thickness. However, we can treat it like a layer that covers the top and vertical sides of the thickness.
    • The exposed sides coming from thickness are:
      • Sides added due to thickness on the top (which will have an area of the top outline) and vertical edges of the platform.
    • The thickness does not change the area of the bottom nor the two faces of the upstanding sections.

So, the total surface area, accounting for the thickness, will be:

  • The original surface area remains and since the part exposed by thickness is vertical (which is generally already covered), we do not have additional top surface area unless specified.

Thus, the surface area of the platform remains: \[ \text{Total Surface Area} = 560 \text{ in}^2 \]

Therefore, the surface area of Nikolai's L-shaped platform is: \[ \boxed{560 \text{ in}^2} \]