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: ?

Explanation:

Step1: Recall the formula for the area of top/bottom (length × width)

Given top and bottom area is \(24\space cm^2\), and from the diagram, length is \(6\space cm\), width is \(x\space cm\)? Wait, no, wait. Wait, the dimensions: let's see, the rectangular prism has length \(6\space cm\), height \(8\space cm\), and width \(x\)? Wait, no, top and bottom area is length × width. Wait, maybe we can find the missing dimension first. Wait, top and bottom area is \(24\space cm^2\), and if length is \(6\space cm\), then width (let's say \(w\)): \(6\times w = 24\), so \(w = 4\space cm\) (since \(24\div6 = 4\)). Then front and back area is \(32\space cm^2\), front/back is length × height. Length is \(6\)? Wait, no, maybe length is \(x\), height is \(8\), width is \(6\)? Wait, maybe I mixed up. Wait, let's re-express. Let the dimensions be length \(l\), width \(w\), height \(h\). Top and bottom area: \(2\times l\times w = 24\), so \(l\times w = 12\)? Wait, no, the problem says "Top and bottom: \(24\space cm^2\)" – so that's the total area of top and bottom, so \(2\times l\times w = 24\), so \(l\times w = 12\). Front and back: total area \(32\space cm^2\), so \(2\times l\times h = 32\), so \(l\times h = 16\). Now, the sides (left and right) are \(2\times w\times h\). We need to find \(w\times h\). From \(l\times w = 12\) and \(l\times h = 16\), we can find \(w = \frac{12}{l}\), \(h = \frac{16}{l}\). Then \(w\times h = \frac{12\times16}{l^2}\)? Wait, no, maybe there's a better way. Wait, looking at the diagram, the base has length \(6\space cm\)? Wait, the diagram shows a rectangular prism with one side \(6\space cm\), height \(8\space cm\), and another side \(x\). Wait, maybe the top and bottom area: if length is \(6\) and width is \(4\) (since \(6\times4 = 24\)? Wait, no, top and bottom total is \(24\), so each is \(12\). Wait, maybe I made a mistake. Wait, let's check the front and back: total \(32\), so each is \(16\). If height is \(8\), then length (the side for front/back) is \(16\div8 = 2\space cm\). Oh! Wait, that makes sense. So front/back area: each is length × height. So if height is \(8\), and front area is \(16\) (since total is \(32\)), then length \(l = 16\div8 = 2\space cm\). Then top and bottom: total area \(24\), so each is \(12\). Top area is length × width, so \(2\times w = 12\), so width \(w = 6\space cm\). Now, the sides (left and right) are width × height, each. So each side is \(6\times8 = 48\)? Wait, no, total sides area is \(2\times w\times h\). So \(w = 6\), \(h = 8\), so \(2\times6\times8 = 96\)? Wait, no, that can't be. Wait, no, wait: length \(l = 2\), width \(w = 6\), height \(h = 8\). Then top and bottom: \(2\times l\times w = 2\times2\times6 = 24\) (correct). Front and back: \(2\times l\times h = 2\times2\times8 = 32\) (correct). Then sides: \(2\times w\times h = 2\times6\times8 = 96\)? Wait, that seems too big. Wait, no, maybe I mixed up length and width. Wait, maybe the base is \(6\) (length) and \(4\) (width), because \(6\times4 = 24\) (top area, but total top and bottom is \(24\)? No, top and bottom total is \(24\), so each is \(12\). So \(l\times w = 12\). Front and back total is \(32\), so \(l\times h = 16\). Then sides: \(w\times h\) × 2. Let's solve for \(w\) and \(h\) in terms of \(l\). \(w = 12/l\), \(h = 16/l\). Then \(w\times h = (12\times16)/l^2\). But we can also find \(l\) from the diagram? Wait, the diagram shows a side with \(6\space cm\), maybe that's the width. Wait, maybe the length is \(4\), width \(6\), height \(8\). Then top and bottom: \(2\times4\times6 = 48\), which is not \(24\). No. Wait, maybe the top and bottom area is \(24\) (total), so \(l\times w = 12\). Front and back total \(32\), so \(l\times h = 16\). Then the sides are \(w\times h\) × 2. Let's find \(w\) and \(h\). Let's assume \(l = 4\), then \(w = 12/4 = 3\), \(h = 16/4 = 4\). Then sides: \(2\times3\times4 = 24\). No. Wait, maybe \(l = 2\), \(w = 6\), \(h = 8\). Then top and bottom: \(2\times2\times6 = 24\) (correct). Front and back: \(2\times2\times8 = 32\) (correct). Then sides: \(2\times6\times8 = 96\). But that seems too big. Wait, maybe the diagram has length \(x = 4\)? Wait, no, the diagram shows a rectangular prism with one edge \(6\space cm\), another \(8\space cm\), and the third \(x\). Wait, maybe the top and bottom area is \(6\times x = 24\) (total? No, top and bottom are two faces, so \(2\times6\times x = 24\), so \(12x = 24\), so \(x = 2\space cm\). Ah! That's it. So length is \(6\), width is \(2\), height is \(8\). Then top and bottom: \(2\times6\times2 = 24\) (correct). Front and back: \(2\times6\times8 = 96\)? No, that's not \(32\). Wait, no, front and back must be \(2\times\) (length × height) or \(2\times\) (width × height)? Wait, no, the front face is usually length × height, and the side face is width × height, and top/bottom is length × width. Wait, maybe the front and back are width × height. Let's redefine: let length \(l = 6\), width \(w = x\), height \(h = 8\). Then top and bottom: \(2\times l\times w = 2\times6\times x = 12x\). The problem says top and bottom is \(24\), so \(12x = 24\) ⇒ \(x = 2\). Then front and back: \(2\times w\times h = 2\times2\times8 = 32\) (correct!). Ah, there we go. So front and back are width × height, not length × height. So now, the sides (left and right) are length × height. So each side is \(l\times h = 6\times8 = 48\), but total sides area is \(2\times l\times h = 2\times6\times8 = 96\)? Wait, no, wait: the sides (the left and right faces) are length × height? Wait, no, the rectangular prism has six faces: top (l×w), bottom (l×w), front (w×h), back (w×h), left (l×h), right (l×h). So total surface area is \(2(lw + wh + lh)\). We know \(lw = 12\) (since \(2lw = 24\)), \(wh = 16\) (since \(2wh = 32\)), so we need to find \(lh\) to get the sides (left and right) total area, which is \(2lh\). Wait, no, the problem is asking "What is the area of the sides?" – the sides here are the left and right faces, which are \(l\times h\) each, so total \(2lh\). We know \(lw = 12\) (from \(2lw = 24\)) and \(wh = 16\) (from \(2wh = 32\)). Let's solve for \(l\) and \(h\) in terms of \(w\). From \(lw = 12\), \(l = 12/w\). From \(wh = 16\), \(h = 16/w\). Then \(lh = (12/w)(16/w) = 192/w^2\). But we can also find \(w\) from the diagram? Wait, no, we already found \(w = 2\) (since \(2lw = 24\) and \(l = 6\)? Wait, no, earlier mistake: if top and bottom is \(24\) (total), then \(lw = 12\) (since \(2lw = 24\)). If the diagram shows length \(l = 6\), then \(w = 12/6 = 2\). Then \(wh = 2\times h = 16\) (since \(2wh = 32\)), so \(h = 8\). Then \(lh = 6\times8 = 48\), so total sides area (left and right) is \(2\times48 = 96\)? Wait, no, that can't be, because then total surface area would be \(24 + 32 + 96 = 152\), but let's check with the formula \(2(lw + wh + lh) = 2(12 + 16 + 48) = 2(76) = 152\). But maybe the problem is using "sides" as the lateral faces excluding top/bottom and front/back? Wait, no, the problem lists top and bottom (24), front and back (32), then sides (?). So sides must be the left and right, which are \(2\times l\times h\). Wait, but with \(l = 6\), \(w = 2\), \(h = 8\), then left and right are \(6\times8\) each, so total \(2\times6\times8 = 96\). But that seems large. Wait, maybe I mixed up length and width. Let's try again: suppose the top and bottom area is \(24\) (total), so \(lw = 12\). Front and back area is \(32\) (total), so \(wh = 16\). Then sides (left and right) area is \(lh\) × 2. Let's find \(l\), \(w\), \(h\). Let's assume \(w = 4\), then \(l = 12/4 = 3\), \(h = 16/4 = 4\). Then sides area is \(2\times3\times4 = 24\). But then front and back would be \(2\times4\times4 = 32\) (correct), top and bottom \(2\times3\times4 = 24\) (correct). Oh! This works. So \(l = 3\), \(w = 4\), \(h = 4\). Then sides (left and right) are \(l\times h = 3\times4 = 12\) each, so total \(24\). Wait, but the diagram shows a height of \(8\space cm\), so \(h = 8\). So maybe the diagram's height is \(8\), so my previous assumption was wrong. Wait, the diagram has a height of \(8\space cm\), so \(h = 8\). Then front and back area: \(2\times w\times h = 32\) ⇒ \(2\times w\times8 = 32\) ⇒ \(16w = 32\) ⇒ \(w = 2\). Then top and bottom area: \(2\times l\times w = 24\) ⇒ \(2\times l\times2 = 24\) ⇒ \(4l = 24\) ⇒ \(l = 6\). Then sides (left and right) area: \(2\times l\times h = 2\times6\times8 = 96\). So that's the area of the sides.

Step2: Calculate the area of the sides

We found that the width \(w = 2\space cm\) (from front/back area: \(2\times w\times8 = 32\) ⇒ \(w = 2\)), length \(l = 6\space cm\) (from top/bottom area: \(2\times l\times2 = 24\) ⇒ \(l = 6\)), height \(h = 8\space cm\). The sides (left and right faces) have area \(2\times l\times h\) (since each side is \(l\times h\) and there are two sides). So:
\[ 2\times6\times8 = 96\space cm^2 \]

Answer:

The area of the sides is \(\boldsymbol{96}\space cm^2\)