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An illustration shows the unfolded version of a rectangular prism composed of 6 rectangles. There are four adjoining rectangles positioned horizontally. The first and third are similar and bigger. The second and fourth are similar and smaller. The area of the second rectangle is labeled 24 square inches. The third rectangle shares the top and bottom sides with two similar rectangles, one on each side. The area of the top rectangle is labeled as 54 square inches and the area of the third rectangle is labeled as 36 square inches.

What is the surface area of the rectangular prism?

(1 point)
in.2
Give it to me in inches

1 answer

To calculate the surface area of the rectangular prism, we first need to identify the dimensions of the rectangles based on the given areas.

  1. Identifying dimensions:

    • The second rectangle (front face) has an area of 24 square inches. We will denote its dimensions as width \( w \) and height \( h_2 \). Therefore, \( w \times h_2 = 24 \).
    • The top rectangle has an area of 54 square inches. Denote its dimensions as width \( l \) and height \( h_1 \). Thus, \( l \times h_1 = 54 \).
    • The third rectangle (back face) has an area of 36 square inches. Denote its dimensions as width \( w \) and height \( h_3 \). Therefore, \( w \times h_3 = 36 \).

  2. Determine dimensions of the rectangles:

    • From the second rectangle: \[ h_2 = \frac{24}{w} \]
    • From the top rectangle: \[ h_1 = \frac{54}{l} \]
    • From the third rectangle: \[ h_3 = \frac{36}{w} \]

  3. Common Dimensions: Since the rectangles are described as adjoining, we can assume the following:

    • The width \( w \) of the second and third rectangles is the same.
    • We can denote \( l \) as the width of the top rectangle.

  4. Height Relations: The height of the given rectangles will help us configure the overall dimensions of the rectangular prism. Using the relationships:

    • The height of the second rectangle \( h_2 \) contributes to the overall height with \( h_1 \) and \( h_3 \).

  5. Surface Area Calculation: The surface area \( SA \) of a rectangular prism can be calculated using the formula: \[ SA = 2(lw + lh + wh) \] Where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

    Now to calculate \( SA \) we need to find \( l \), \( w \), and \( h \).

    • Using the known dimensions:
      • We have \( w \) from the equations above.
      • We can set configurations based upon adjacent rectangles contributing to the dimensions.

  6. Assuming common values to facilitate calculation:

    • From the area of rectangle 2: let’s assume \( w = 6 \) (since \( w \times h_2 = 24 \), then \( h_2 = 4 \)).
    • Then verify height for \( h_3 \): \( h_3 = \frac{36}{6} = 6 \).
    • So configurations suggest:
      • \( l = 9 \), (with \( h_1 = 6 \)).

Thus, we find: \[ SA = 2(lw + lh + wh) = 2(69 + 94 + 6*4) \]

Calculating:

  • Volume contribution to each side: \[ lw = 54 \] \[ lh = 36 \] \[ wh = 24 \]

Calculating: \[ SA = 2(54 + 36 + 24) = 2 \cdot 114 = 228 \text{ square inches} \]

Thus the total surface area of the rectangular prism is:

\[ \boxed{228} \text{ square inches.} \]