Question

1 cm 6 cm 4 cm 6 cm 1 cm find the surface area of the prism using the net. what is the area of the top and bottom rectangles? front and back: 8 cm² rectangles side: 12 cm² rectangles top and bottom: ? cm² rectangles surface area: cm²

Explanation:

Step1: Identify dimensions of top/bottom

The top and bottom rectangles have length \( 4 \, \text{cm} \) and width \( 1 \, \text{cm} \)? Wait, no, looking at the net, the top and bottom (from the prism's net) – wait, actually, the top and bottom rectangles: let's check the dimensions. Wait, the front/back: maybe I misread. Wait, the prism's net: the purple rectangles are 6cm (height) and 4cm? Wait, no, the top and bottom: let's see, the top and bottom rectangles – from the net, the top and bottom (the small rectangles? Wait, no, the problem is asking for top and bottom rectangles. Wait, the prism is a rectangular prism. Let's find the length and width of the top/bottom. From the net, the top and bottom rectangles: looking at the net, the top and bottom (the two rectangles that are top and bottom) – their dimensions: length is \( 4 \, \text{cm} \) and width is \( 1 \, \text{cm} \)? Wait, no, maybe the top and bottom are the rectangles with length \( 4 \, \text{cm} \) and width \( 1 \, \text{cm} \)? Wait, no, wait the front and back are 8 cm², side are 12 cm². Wait, let's re-express.

Wait, the formula for the area of a rectangle is \( \text{length} \times \text{width} \). For the top and bottom rectangles: let's find their length and width. From the net, the top and bottom rectangles – looking at the diagram, the top and bottom (the two rectangles at the top and bottom of the central purple rectangle) – their dimensions: length is \( 4 \, \text{cm} \) and width is \( 1 \, \text{cm} \)? Wait, no, maybe the top and bottom are the rectangles with length \( 4 \, \text{cm} \) and width \( 1 \, \text{cm} \), but wait, no, let's check the front and back. Front and back are 8 cm², so each front/back is \( 8 \div 2 = 4 \, \text{cm}^2 \). Side rectangles: 12 cm², so each side is \( 12 \div 2 = 6 \, \text{cm}^2 \). Now, for top and bottom: let's find the dimensions. The prism has length, width, height. Let's assume: height is 6 cm (from the purple rectangles), length is 4 cm, width is 1 cm. Wait, no, the top and bottom rectangles: length is 4 cm, width is 1 cm? Wait, no, the area of one top/bottom rectangle would be \( 4 \times 1 = 4 \, \text{cm}^2 \), but since there are two (top and bottom), total area is \( 2 \times (4 \times 1) = 8 \, \text{cm}^2 \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, looking at the net, the top and bottom rectangles: the length is 6 cm? No, the purple rectangles are 6 cm (height) and 4 cm (width). Wait, the top and bottom: the small rectangles at the top and bottom of the central purple rectangle – their length is 4 cm and width is 1 cm. So area of one top/bottom rectangle is \( 4 \times 1 = 4 \, \text{cm}^2 \), so two of them (top and bottom) would be \( 2 \times 4 = 8 \, \text{cm}^2 \)? Wait, but let's check the surface area. Wait, the front and back: 8 cm² (so two rectangles, each 4 cm²), side: 12 cm² (two rectangles, each 6 cm²), top and bottom: let's recalculate. Wait, maybe the top and bottom rectangles have dimensions 6 cm and 1 cm? No, 6×1=6, two of them would be 12. Wait, no, the diagram shows 1 cm, 4 cm, 6 cm. Let's list the faces:

- Front/back: two rectangles, each with area \( 4 \times 2 \)? Wait, no, the front and back are 8 cm² total, so each is 4 cm². So if front/back area is 4 cm², then dimensions are, say, 4 cm (length) and 1 cm (height)? No, 4×1=4. Then side rectangles: total 12 cm², so each is 6 cm². So dimensions 6 cm (height) and 1 cm (width)? 6×1=6. Then top and bottom: dimensions 4 cm (length) and 6 cm (width)? Wait, 4×6=24, two of them would be 48. That can't be. Wait, I think I misread the diagram. Let's look again: the purple rectangles are 6 cm (height) and 4 cm (width). The small rectangles (top, bottom, left, right) are 1 cm (width) and 4 cm (length) for top/bottom, and 1 cm (width) and 6 cm (height) for left/right. Wait, so:

- Front/back: purple rectangles, area \( 6 \times 4 = 24 \, \text{cm}^2 \) each? But the problem says front and back total 8 cm². Oh! Wait, the problem's text says "Front and back: 8 cm² rectangles" – so total front and back is 8 cm². So each front/back is 4 cm². So dimensions: let's say front/back is 4 cm², so length × height = 4. Then side rectangles: total 12 cm², so each side is 6 cm², so length × width = 6? Wait, no, the prism is a rectangular prism, so surface area is \( 2(lw + lh + wh) \), where \( l \) is length, \( w \) is width, \( h \) is height.

Given:

- Front/back: \( 2lh = 8 \) ⇒ \( lh = 4 \)

- Side: \( 2wh = 12 \) ⇒ \( wh = 6 \)

- Top/bottom: \( 2lw \) (we need to find this)

We need to find \( 2lw \). Let's find \( l \), \( w \), \( h \). From \( lh = 4 \) and \( wh = 6 \), divide the two equations: \( \frac{lh}{wh} = \frac{4}{6} \) ⇒ \( \frac{l}{w} = \frac{2}{3} \) ⇒ \( l = \frac{2}{3}w \). But maybe from the diagram, the dimensions are: looking at the net, the small rectangles (top, bottom) have length 4 cm and width 1 cm? Wait, the diagram shows 1 cm, 4 cm, 6 cm. Wait, the top and bottom rectangles: length is 4 cm, width is 1 cm. So area of one is \( 4 \times 1 = 4 \, \text{cm}^2 \), so two (top and bottom) is \( 2 \times 4 = 8 \, \text{cm}^2 \)? But then surface area would be 8 (front/back) + 12 (side) + 8 (top/bottom) = 28? But let's check with the formula. Wait, maybe the front/back is 8 cm² (total), side is 12 cm² (total), top/bottom is? Wait, the diagram's net: the purple rectangles are 6 cm (height) and 4 cm (width). The small rectangles (top, bottom, left, right) are 1 cm (width) and 4 cm (length) for top/bottom, and 1 cm (width) and 6 cm (height) for left/right. Wait, so:

- Front/back: two purple rectangles, each \( 6 \times 4 = 24 \), but the problem says front/back is 8. So I must have misread. Wait, the problem's text: "Front and back: 8 cm² rectangles" – so total front and back is 8. So each is 4. So dimensions: 4 cm² per front/back, so length × height = 4. Then side: 12 cm² total, so each side is 6, so width × height = 6. Then top/bottom: length × width × 2. Let's find length, width, height. Let's assume height is 2 cm. Then length = 4 / 2 = 2 cm, width = 6 / 2 = 3 cm. Then top/bottom area is 2×3×2=12. But that doesn't match. Wait, maybe the diagram's 1 cm, 4 cm, 6 cm: the top and bottom rectangles have length 4 cm and width 1 cm. So area of one is 4×1=4, two is 8. Then surface area: 8 (front/back) + 12 (side) + 8 (top/bottom) = 28. But let's check the net. The net has two purple rectangles (front/back) with 6 cm and 4 cm? No, the 6 cm is the height, 4 cm is the length, 1 cm is the width. Wait, maybe the front/back are the small rectangles? No, the purple ones are larger. Wait, the problem is asking "What is the area of the top and bottom rectangles?" So let's look at the net: the top and bottom rectangles (the two small rectangles at the top and bottom of the central purple rectangle) – their dimensions are 4 cm (length) and 1 cm (width). So area of one is 4×1=4, so two (top and bottom) is 4×2=8. Then surface area is 8 (front/back) + 12 (side) + 8 (top/bottom) = 28. But let's confirm with the formula. The surface area of a rectangular prism is \( 2(lw + lh + wh) \). If l=4, w=1, h=6: then \( 2(4×1 + 4×6 + 1×6) = 2(4 + 24 + 6) = 2×34=68 \). No, that's not. Wait, maybe l=4, w=1, h=2: then \( 2(4×1 + 4×2 + 1×2)=2(4+8+2)=28 \). Ah! So h=2. Then front/back: 2×(4×2)=16? No, the problem says front/back is 8. So 2×(l×h)=8 ⇒ l×h=4. If l=4, h=1: then l×h=4. Then side: 2×(w×h)=12 ⇒ w×h=6. If h=1, then w=6. Then top/bottom: 2×(l×w)=2×(4×6)=48. Then surface area: 8+12+48=68. But the diagram shows 6 cm. So I think the diagram's 6 cm is the height, 4 cm is length, 1 cm is width. Then front/back: 2×(4×1)=8 (matches), side: 2×(6×1)=12 (matches), top/bottom: 2×(4×6)=48. Oh! Wait, that makes sense. So front/back: each is 4×1=4, two of them: 8. Side: each is 6×1=6, two of them: 12. Top/bottom: each is 4×6=24, two of them: 48. Then surface area is 8+12+48=68. So the area of top and bottom rectangles is 48 cm².

Wait, I see my mistake earlier. The front/back rectangles: length 4, width 1 (so area 4×1=4 each, two of them 8). Side rectangles: length 6, width 1 (area 6×1=6 each, two of them 12). Top/bottom rectangles: length 4, width 6 (area 4×6=24 each, two of them 48). So that's correct. So the area of top and bottom rectangles is 48 cm².

Step1: Determine dimensions of top/bottom

The top and bottom rectangles have length \( 4 \, \text{cm} \) and width \( 6 \, \text{cm} \) (from the net: the top/bottom faces correspond to the length of the front/back (4 cm) and the height of the side faces (6 cm)).

Step2: Calculate area of one top/bottom

Area of one rectangle: \( \text{length} \times \text{width} = 4 \times 6 = 24 \, \text{cm}^2 \).

Step3: Calculate total for top and bottom

Since there are two (top and bottom), total area: \( 2 \times 24 = 48 \, \text{cm}^2 \).

Answer:

48