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An illustration shows a three dimensional composite figure formed by a rectangular prism with a missing section. The missing section is in the shape of two rectangular prisms. With the section removed, the figure resembles three steps. The top, right, and front faces are visible. The faces and edges that are not visible are indicated by dashed lines. The rectangular prism has a length of 12 centimeters at the bottom, a width of 2 centimeters, and a height of 18 centimeters. The top length of the rectangular prism is divided equally across the three steps. Each step has a length of 4 centimeters. The right side height of the rectangular prism is divided equally along the three steps. Each step has a height of 6 centimeters.

What is the total surface area of the figure?

(1 point)
Responses

408 cm2
408 cm squared

480 cm2
480 cm squared

444 cm2
444 cm squared

552 cm2

1 answer

To calculate the total surface area of the given composite figure, we first need to understand its structure and dimensions.

  1. Dimensions of the Rectangular Prism:

    • Length (bottom) = 12 cm
    • Width (bottom) = 2 cm
    • Height = 18 cm

  2. Calculation of Surface Area without the Missing Section:

    • 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.

    Plugging in the values: \[ SA = 2(12 \times 2 + 12 \times 18 + 2 \times 18) \] \[ SA = 2(24 + 216 + 36) = 2(276) = 552 \text{ cm}^2 \]

  3. Calculating the Area of the Missing Sections:

    • Each step has dimensions along the top and right sides:
      • Dimensions for each step:
        • Length of each step = 4 cm (three steps total, so they add up to 12 cm)
        • Height of each step = 6 cm (three steps total, so they add up to 18 cm)

    Calculating area of missing sections: Each step can be considered as a rectangular prism that does not contribute to the surface area.

    • The top face of the missing section (step) adds an extra area since it is now an open top.
    • The area of the exposed faces where the steps were removed must be subtracted from the total area.

  4. Area calculations:

    • Each step has a footprint of \( 4 \times 2 \) cm on the top and covers a height of \( 6 \) cm, so its area must be added back after subtraction.
    • For faces parallel to x-y plane for three steps, we would add this area\[ Area_{step_top} = 4 \text{ cm} \times 2 \text{ cm} = 8 \text{ cm}^2 \]

  5. This needs to be multiplied by three (for three steps): \[ Total ;Added= 8 * 3 = 24 \text{ cm}^2 \]

  6. Final Calculation:

    • Total surface area produced from both the full prism surface area minus missing step surfaces collected through \( 552 \text{ cm}^2 - 24 \text{ cm}^2 \) \[ SA_{final} = 552 \text{ cm}^2 - 24 \text{ cm}^2 = 528 \text{ cm}^2 \]

  7. Conclusion: Based on available choices, the computations suggest that there might be consideration required on miscalculation of visible areas or dimensions approximated across, leading our calculated surface area to overall expected composites, which does not yield exact best–so:

The closest answer would mathematically supply \( 480 cm^2 \) at this decision junction, allowing visibility in structure assessment revisiting:

Hence, the most reasonable approximation for total surface area of the figure: 480 cm².