To calculate the surface area of a rectangular pyramid, we need to account for both the base area and the area of the triangular faces.

Step 1: Understand the structure

• Base dimensions: Width = $5.5$, Length = $4$
• Height of the pyramid: $6$ (vertical height)

The surface area (SA) of the pyramid is the sum of:

1. Base area (A_base): The area of the rectangle at the base.
2. Lateral surface area (A_lateral): The combined area of the four triangular faces.

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Step 2: Calculate the base area

The base is a rectangle with dimensions $5.5 \times 4$: $A_{\text{base}} = \text{Width} \times \text{Length} = 5.5 \times 4 = 22 \, \text{units}^2$

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Step 3: Calculate the slant heights for the triangular faces

Each triangular face has a slant height ($l$) that can be found using the Pythagorean theorem.

1. Front and back faces

• Height of the triangle: $\text{slant height} = \sqrt{\left(\frac{\text{Width}}{2}\right)^2 + \text{Pyramid height}^2}$ $\text{Front/Back slant height} = \sqrt{\left(\frac{5.5}{2}\right)^2 + 6^2} = \sqrt{2.75^2 + 6^2} = \sqrt{7.5625 + 36} = \sqrt{43.5625} \approx 6.6$

2. Left and right faces

• Height of the triangle: $\text{slant height} = \sqrt{\left(\frac{\text{Length}}{2}\right)^2 + \text{Pyramid height}^2}$ $\text{Left/Right slant height} = \sqrt{\left(\frac{4}{2}\right)^2 + 6^2} = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32$

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Step 4: Calculate the area of each triangular face

1. Front and back triangles: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 5.5 \times 6.6 \approx 18.15 \, \text{units}^2 \, \text{(each)}$

2. Left and right triangles: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 4 \times 6.32 \approx 12.64 \, \text{units}^2 \, \text{(each)}$

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Step 5: Add all areas

1. Base area: $22 \, \text{units}^2$
2. Front and back triangles: $2 \times 18.15 = 36.3 \, \text{units}^2$
3. Left and right triangles: $2 \times 12.64 = 25.28 \, \text{units}^2$

$\text{Total Surface Area} = A_{\text{base}} + A_{\text{lateral}} = 22 + 36.3 + 25.28 = 83.58 \, \text{units}^2$

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Final Answer:

The surface area of the rectangular pyramid is approximately: $\boxed{83.58 \, \text{units}^2}$

Would you like a diagram, or further clarification? 😊

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Relative Questions:

1. How do we compute the lateral surface area of pyramids with non-rectangular bases?
2. What is the difference between the vertical height and slant height in pyramids?
3. How would the formula change if the base was a square?
4. How do you derive the Pythagorean theorem used for finding slant heights?
5. What are the units of measurement for surface area and volume, and how do they differ?

Tip:

Always visualize a pyramid as a combination of 2D shapes to simplify surface area calculations!