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Break down the composite figure into three figures. The bottom rectangle has dimensions 15×6×6. The top side rectangles have dimensions 4×6×6. Remember to subtract the hidden faces from your total surface area. The area to subtract includes the left side bottom of the top rectangle and top of the bottom rectangle. It also includes the right side bottom of the top rectangle and top of the bottom rectangle

1 answer

To find the total surface area of the composite figure, we need to break it down into its constituent parts, calculate the surface area of each part, and then subtract the areas of the hidden faces.

  1. Identify the Figures:
    • Bottom rectangle (Cuboid): Dimensions = 15 (length) × 6 (width) × 6 (height).
    • Top rectangles (Cuboid): Dimensions = 4 (length) × 6 (width) × 6 (height). Since there are two top rectangles, we will consider their collective dimensions separately.

Step 1: Calculate Surface Area of Each Figure

Bottom Rectangle (Large Cuboid)

  • Surface area formula for a cuboid: \( SA = 2(lw + lh + wh) \)
  • Dimensions: \( l = 15 \), \( w = 6 \), \( h = 6 \)

Calculating:

\[ SA_{\text{bottom}} = 2(15 \times 6 + 15 \times 6 + 6 \times 6) = 2(90 + 90 + 36) = 2 \times 216 = 432 \]

Top Rectangles (Small Cuboid)

  • Each top rectangle has the same dimensions: \( l = 4 \), \( w = 6 \), \( h = 6 \)

Calculating for one:

\[ SA_{\text{top}} = 2(4 \times 6 + 4 \times 6 + 6 \times 6) = 2(24 + 24 + 36) = 2 \times 84 = 168 \]

Since there are two top rectangles:

\[ SA_{\text{tops}} = 2 \times 168 = 336 \]

Step 2: Calculate Total Surface Area Before Subtraction

\[ SA_{\text{total}} = SA_{\text{bottom}} + SA_{\text{tops}} = 432 + 336 = 768 \]

Step 3: Hidden Faces Area to Subtract

Each pair of top rectangles hides two areas on the bottom rectangle. The hidden areas are:

  • The area of the left side bottom of the top rectangle which touches the top face of the bottom rectangle (4 x 6).
  • The area of the right side bottom of the top rectangle which also touches the top face of the bottom rectangle (4 x 6).

Calculating the hidden areas:

\[ \text{Hidden Area (left side)} = 4 \times 6 = 24 \] \[ \text{Hidden Area (right side)} = 4 \times 6 = 24 \]

Total hidden area to subtract:

\[ \text{Total Hidden Area} = 24 + 24 = 48 \]

Step 4: Calculate Final Surface Area

\[ SA_{\text{final}} = SA_{\text{total}} - \text{Total Hidden Area} = 768 - 48 = 720 \]

Conclusion

The total surface area of the composite figure, after subtracting the hidden faces, is 720 square units.