Question

follow the steps to find the surface area of the rectangular prism. what is the area of the front and back? top and bottom: 24 cm² front and back: ? sides: total sa: diagrams of rectangular prisms with dimensions 2 cm, 6 cm, 8 cm

Explanation:

Step1: Identify front/back dimensions

Front/back: height = 8 cm, width = 2 cm? Wait, no—wait, top/bottom area is 24 cm². Top/bottom: length × width. Given length 6 cm? Wait, the prism has length 2 cm, width 6 cm? Wait, no, the diagram: base is 2 cm (length) and 6 cm (width)? Wait, no, top and bottom area is 24 cm². Let's check: top/bottom area = length × width = 24. From the diagram, the base has 2 cm and 6 cm? Wait, 2×6=12, no. Wait, maybe length is 6, width is 4? No, wait the front/back: height is 8 cm, and the other dimension—wait, the front face: height 8 cm, and the vertical side? Wait, no, let's re-express.

Wait, the rectangular prism has length \( l = 2 \) cm, width \( w = 6 \) cm? No, top and bottom area is 24 cm². So \( l \times w = 24 \). Wait, the diagram shows one side as 2 cm, another as 6 cm, height 8 cm. Wait, maybe length is 6, width is 4? No, 2×12=24? Wait, no, the front and back faces: their area is height × length (or height × width). Wait, the front face: height is 8 cm, and the side is 2 cm? No, wait the prism's dimensions: let's see, the base is 2 cm (depth) and 6 cm (width), height 8 cm? Wait, no, top and bottom area: if top is length × width, and front is height × width. Wait, maybe the top/bottom area is length × width = 24, so length × width =24. From the diagram, one side is 2 cm (maybe length), so width would be 24 / 2 =12? No, that doesn't match. Wait, no, the diagram has 2 cm, 6 cm, 8 cm. Wait, 2×6=12, 2×8=16, 6×8=48. Wait, top and bottom: maybe length 6, width 4? No, the given top and bottom is 24. So 6×4=24? But the diagram shows 2,6,8. Wait, maybe I misread. Wait, the problem says "Top and bottom: 24 cm²". So top area is length × width =24. Then front and back: each is height × length (or height × width). Wait, the prism's height is 8 cm, and the length (for front) is 2 cm? No, wait, let's check the dimensions: the base has 2 cm (let's say length) and 6 cm (width), height 8 cm. Then top area: 2×6=12, but the problem says top is 24. So maybe the base is 6 cm (length) and 4 cm (width), but the diagram shows 2,6,8. Wait, maybe the top and bottom area is length × width =24, so length=6, width=4? No, the diagram has 2,6,8. Wait, perhaps the 2 cm is the width, 6 cm is the length, and height 8 cm. Then top area: 6×2=12, but the problem says 24. So maybe the top and bottom area is 24, so length × width =24. So if length is 6, width is 4 (6×4=24), but the diagram shows 2,6,8. Wait, maybe the 2 cm is the width, 8 cm is the length? No, 2×8=16≠24. Wait, 3×8=24? No, the diagram has 2,6,8. Wait, maybe the top and bottom area is 24, so that's length × width =24. Then front and back: each is height × length. The height is 8 cm, length is 6 cm? No, 8×6=48, but that's too big. Wait, no, maybe the front face is height × width. The width is 2 cm, height 8 cm: 2×8=16, but that's not matching. Wait, maybe I made a mistake. Wait, the problem is to find the area of front and back. Let's recall: a rectangular prism has 6 faces: top/bottom (2 faces), front/back (2 faces), left/right (2 faces). The formula for front/back area: 2 × (height × length) or 2 × (height × width), depending on orientation. Wait, the given top and bottom area is 24 cm², which is 2 × (length × width) =24? No, top and bottom are two faces, so each is length × width, so total top and bottom area is 2×(l×w)=24? Wait, no, the problem says "Top and bottom: 24 cm²"—maybe that's the total for both? So each top and bottom is 12 cm²? No, the problem says "Top and bottom: 24 cm²" as a total? Wait, no, maybe "Top and bottom" refers to the area of one top and one bottom, so total 24. So l×w=24? Wait, the diagram shows the base with 2 cm and 6 cm. Wait, 2×6=12, so if top and bottom total 24, that would be 2×(2×6)=24. Ah! Yes, that makes sense. So top and bottom: each is 2×6=12, so two of them: 2×12=24. So length l=2 cm, width w=6 cm, height h=8 cm. Then front and back: each is height × length (h×l), so two faces: 2×(h×l). Wait, h=8, l=2: 2×(8×2)=32? No, wait, front face: if the front is height × width? Wait, no, the front face: if the base is length=2, width=6, then the front face would be height × width? Wait, no, let's visualize: the prism has length (depth) 2 cm, width 6 cm, height 8 cm. So the front and back faces are the ones with height 8 and width 6? Wait, no, that would be the sides. Wait, I'm confused. Wait, let's use the formula for surface area of a rectangular prism: \( SA = 2(lw + lh + wh) \). We know that \( lw \) (top/bottom) total is 24? Wait, no, \( 2lw = 24 \), so \( lw = 12 \). From the diagram, l=2, w=6: 2×6=12, so that's correct. Then \( lh \) is front/back: each front face is l×h, so two faces: 2×(l×h). l=2, h=8: 2×(2×8)=32? Wait, no, 2×8=16 per face, two faces: 32. Wait, but let's check: l=2, w=6, h=8. Then:

- Top/bottom: 2×(l×w) = 2×(2×6)=24 (matches)
- Front/back: 2×(l×h) = 2×(2×8)=32
- Sides: 2×(w×h)=2×(6×8)=96
- Total SA: 24+32+96=152

But the problem is asking for front and back area. So using l=2, h=8, two faces: 2×(2×8)=32. Wait, but let's confirm the dimensions. The diagram shows the base as 2 cm (length) and 6 cm (width), height 8 cm. So front face: length=2, height=8, so area per front face is 2×8=16, two faces (front and back) is 2×16=32.

Step1: Determine front/back dimensions

Front/back faces: height \( h = 8 \) cm, length \( l = 2 \) cm (from diagram). Each face area: \( l \times h \).

Step2: Calculate total front/back area

Total for two faces: \( 2 \times (l \times h) = 2 \times (2 \times 8) \).

Step3: Compute the value

\( 2 \times 16 = 32 \) cm².

Answer:

32 cm²