Question

follow the steps to find the surface area of the rectangular prism. what is the area of the sides? top and bottom: 24 cm² front and back: 32 cm² sides:? total sa: 8 cm 4 cm 2 cm

Explanation:

Step1: Identify the dimensions

The rectangular prism has length \( l = 4 \, \text{cm} \), width \( w = 2 \, \text{cm} \), and height \( h = 8 \, \text{cm} \). The sides (left and right faces) are rectangles with dimensions width and height, i.e., \( w = 2 \, \text{cm} \) and \( h = 8 \, \text{cm} \)? Wait, no, wait. Wait, the top and bottom area is \( 24 \, \text{cm}^2 \), let's check: top and bottom are length times width. If length is \( 4 \) and width is \( 2 \), \( 4\times2 = 8 \), no. Wait, maybe length is \( 6 \)? Wait, no, the diagram shows length \( 4 \, \text{cm} \), width \( 2 \, \text{cm} \), height \( 8 \, \text{cm} \)? Wait, no, the front and back area is \( 32 \, \text{cm}^2 \). Front and back are length times height. If front area is \( 32 \), and height is \( 8 \), then length is \( 32\div8 = 4 \, \text{cm} \). Top and bottom area is \( 24 \, \text{cm}^2 \), top and bottom are length times width, so \( 4\times w = 24 \)? No, \( 24\div4 = 6 \), so width is \( 6 \)? Wait, maybe I misread the diagram. Wait, the sides (left and right) are width times height. Wait, let's use the formula for the area of the sides (lateral faces: left and right). The formula for the area of the sides (left and right) of a rectangular prism is \( 2\times(\text{width}\times\text{height}) \)? Wait, no, the three pairs of faces: top/bottom (length×width), front/back (length×height), left/right (width×height). We know top/bottom area is \( 24 \, \text{cm}^2 \), front/back is \( 32 \, \text{cm}^2 \). Let's denote:

Let length \( l \), width \( w \), height \( h \).

Top/bottom area: \( 2(lw) = 24 \) ⇒ \( lw = 12 \).

Front/back area: \( 2(lh) = 32 \) ⇒ \( lh = 16 \).

We need to find the area of the sides (left/right), which is \( 2(wh) \).

From \( lw = 12 \) and \( lh = 16 \), divide the two equations: \( \frac{lh}{lw} = \frac{16}{12} \) ⇒ \( \frac{h}{w} = \frac{4}{3} \) ⇒ \( h = \frac{4}{3}w \).

But maybe easier: let's find \( w \) and \( h \). From \( lw = 12 \) and \( lh = 16 \), let's assume \( l = 4 \) (from front area: \( 4\times h = 16 \) ⇒ \( h = 4 \)? No, front area is \( 32 \), so \( l\times h = 16 \)? Wait, no, the front and back area is \( 32 \, \text{cm}^2 \), so each front face is \( 16 \, \text{cm}^2 \). So \( l\times h = 16 \). Top and bottom area is \( 24 \, \text{cm}^2 \), so each top face is \( 12 \, \text{cm}^2 \), so \( l\times w = 12 \). Then, if we divide \( lh \) by \( lw \), we get \( h/w = 16/12 = 4/3 \), so \( h = (4/3)w \). But maybe the dimensions are \( l = 4 \), \( w = 3 \), \( h = 4 \)? No, that doesn't make sense. Wait, maybe the diagram has length \( 4 \), width \( 3 \), height \( 8 \)? No, front area is \( 32 \), so \( 4\times8 = 32 \), yes! So length \( l = 4 \), height \( h = 8 \), so front area is \( 4\times8 = 32 \), which matches. Then top and bottom area: \( l\times w = 24 \), so \( 4\times w = 24 \) ⇒ \( w = 6 \)? Wait, no, \( 4\times6 = 24 \), yes. So length \( l = 4 \), width \( w = 6 \), height \( h = 8 \)? But the diagram shows width as \( 2 \)? Wait, maybe I misread the diagram. Wait, the diagram has a prism with length \( 4 \, \text{cm} \), width \( 2 \, \text{cm} \), height \( 8 \, \text{cm} \)? No, the numbers on the diagram: 4 cm (length), 2 cm (width), 8 cm (height). Wait, top and bottom area: length×width = 4×2 = 8, but it's given as 24. So maybe the length is 6, width 4, height 8? No, front area is 32, 8×4=32, yes. Then top and bottom area: 6×4=24, yes! So length \( l = 6 \), width \( w = 4 \), height \( h = 8 \)? No, the diagram shows 4 cm, 2 cm, 8 cm. Wait, maybe the diagram has length 4, width 3, height 8? No, this is confusing. Wait, let's use the given areas. Top and bottom: 24, so each is 12. Front and back: 32, so each is 16. The sides (left and right) are each width×height. Let's denote width as \( w \), height as \( h \). We know that length×width = 12 (from top/bottom: 24 total, so 12 each), and length×height = 16 (from front/back: 32 total, so 16 each). So length is the same, so \( \frac{length\times height}{length\times width} = \frac{16}{12} \) ⇒ \( \frac{h}{w} = \frac{4}{3} \) ⇒ \( h = \frac{4}{3}w \). Now, the area of one side (left or right) is \( w\times h = w\times\frac{4}{3}w = \frac{4}{3}w^2 \). But we need another way. Wait, maybe the prism has length 4, width 3, height 8? No, let's look at the diagram again. The prism has a base with length 4 cm, width 2 cm, height 8 cm? No, the numbers on the diagram: 4 cm (length), 2 cm (width), 8 cm (height). Wait, maybe the top and bottom area is 24, so 4×6=24, so width is 6? Maybe the diagram's width is 6, length 4, height 8. Then front area is 4×8=32, which matches. Then the sides (left and right) are width×height = 6×8=48? No, that can't be. Wait, no, the sides (left and right) are width×height? No, left and right are width×height? Wait, no, the three dimensions: length (l), width (w), height (h). The faces:

- Top/bottom: l × w (area: lw each, total 2lw)
- Front/back: l × h (area: lh each, total 2lh)
- Left/right: w × h (area: wh each, total 2wh)

We are given:

- Total top/bottom area: 2lw = 24 ⇒ lw = 12
- Total front/back area: 2lh = 32 ⇒ lh = 16
- We need to find total left/right area: 2wh

To find 2wh, we can multiply lw and lh: (lw)(lh) = l²wh = 12×16 = 192. But we don't know l. Alternatively, divide lh by lw: (lh)/(lw) = h/w = 16/12 = 4/3 ⇒ h = (4/3)w. Then substitute into lw = 12: l×w = 12 ⇒ l = 12/w. Then lh = (12/w)×(4/3)w = 16, which checks out. Now, we need to find wh. From h = (4/3)w, wh = w×(4/3)w = (4/3)w². But we need another approach. Wait, maybe the length is 4, so from lh = 16, h = 16/4 = 4. Then from lw = 12, w = 12/4 = 3. So width is 3, height is 4. Then the area of the sides (left and right) is 2×(w×h) = 2×(3×4) = 24? Wait, no, 3×4=12, times 2 is 24? But let's check with the diagram. Wait, the diagram shows length 4, width 2, height 8. Maybe I made a mistake. Wait, let's use the given dimensions in the diagram: length 4 cm, width 2 cm, height 8 cm. Then top/bottom area: 4×2×2=16, but it's given as 24. So that's not matching. So maybe the length is 6, width 4, height 8. Then top/bottom: 6×4×2=48, no. Wait, the problem says "the area of the sides". The sides are the left and right faces, which are width×height. Wait, maybe the width is 3, height is 8? No, let's look at the numbers given: top and bottom 24, front and back 32. Let's find the area of the sides. The total surface area of a rectangular prism is 2(lw + lh + wh). We know 2lw = 24, 2lh = 32, so we need to find 2wh. Let's denote S = 2(lw + lh + wh) = 24 + 32 + 2wh. But we can find wh by finding w and h. From 2lw = 24 ⇒ lw = 12. From 2lh = 32 ⇒ lh = 16. Let's solve for w and h. Let l be a common factor. Let l = 4, then w = 12/4 = 3, h = 16/4 = 4. Then wh = 3×4 = 12, so 2wh = 24. So the area of the sides (left and right) is 24 cm²? Wait, no, 2wh is the total area of both sides. Wait, the problem says "the area of the sides" – maybe it's the total area of the two sides (left and right), so 2wh. So if wh = 12, then 2wh = 24. Let's check: total surface area would be 24 (top/bottom) + 32 (front/back) + 24 (sides) = 80 cm². Let's verify with the formula: 2(lw + lh + wh) = 2(12 + 16 + 12) = 2(40) = 80. Yes, that works. So the area of the sides is 24 cm², and total surface area is 80 cm².

Step2: Calculate the area of the sides

We know that the total surface area formula is \( 2(lw + lh + wh) \). We are given \( 2lw = 24 \) (top and bottom) and \( 2lh = 32 \) (front and back). Let \( A_{\text{sides}} = 2wh \) (area of left and right sides). Then:

\( 2(lw + lh + wh) = 24 + 32 + A_{\text{sides}} \)

But also, \( 2(lw + lh + wh) = 2lw + 2lh + 2wh = 24 + 32 + 2wh \).

We can find \( wh \) by first finding \( l \), \( w \), \( h \). From \( 2lw = 24 \), \( lw = 12 \); from \( 2lh = 32 \), \( lh = 16 \). Let's assume \( l = 4 \) (since \( 16 \div 4 = 4 \) and \( 12 \div 4 = 3 \)), so \( w = 3 \), \( h = 4 \). Then \( wh = 3 \times 4 = 12 \), so \( 2wh = 24 \). Thus, the area of the sides is \( 24 \, \text{cm}^2 \).

Step3: Calculate the total surface area

Total surface area (SA) is the sum of the areas of all faces: top/bottom (24) + front/back (32) + sides (24) = \( 24 + 32 + 24 = 80 \, \text{cm}^2 \).

Answer:

The area of the sides is \( \boldsymbol{24 \, \text{cm}^2} \) and the total surface area is \( \boldsymbol{80 \, \text{cm}^2} \).