Question

choose two figures that have the same surface area but different volumes. select the correct choices. show your reasoning.

Explanation:

Step1: Analyze Figure C

Assume each small cube has side length 1. Figure C: dimensions \(3\times3\times1\). Surface area formula for rectangular prism: \(2(lw + lh + wh)\). So \(2(3\times3 + 3\times1 + 3\times1)=2(9 + 3 + 3)=2\times15 = 30\). Volume: \(3\times3\times1 = 9\).

Step2: Analyze Figure D

Figure D: dimensions \(6\times1\times1\). Surface area: \(2(6\times1 + 6\times1 + 1\times1)=2(6 + 6 + 1)=2\times13 = 26\)? Wait, no, maybe I miscounted. Wait, maybe Figure C is \(3\times2\times1\)? Wait, maybe better to count faces. Alternatively, Figure C: let's count cubes. If Figure C has 9 cubes (3x3x1), Figure D has 6 cubes (6x1x1). Wait, maybe another approach. Wait, maybe Figure E: let's see, Figure E: let's count layers. Suppose Figure E has 7 cubes? Wait, maybe the correct pair is C and E? Wait, no, let's re - evaluate.

Wait, maybe Figure C: length = 3, width = 3, height = 1. Surface area: \(2(3\times3 + 3\times1 + 3\times1)=30\). Volume: \(9\). Figure E: let's count the cubes. Let's say it's a 2x3x1? No, looking at the figure, Figure E has a column with 3 cubes and another with 4? Wait, maybe I made a mistake. Alternatively, Figure C (volume 9, surface area 30) and Figure D (volume 6, surface area: let's calculate again. Figure D: 6 cubes in a line (1x1x6). Surface area: \(2(1\times1 + 1\times6 + 1\times6)=2(1 + 6 + 6)=26\). No, that's not same. Wait, maybe Figure C and Figure E. Let's assume Figure E has dimensions that give surface area 30. Suppose Figure E has 7 cubes. Volume 7, surface area: let's calculate. If it's a 3x2x1 with one cube on top, maybe. Alternatively, the correct pair is C and D? No, volume 9 vs 6, surface area 30 vs 26. Wait, maybe I misread the figures. Let's assume the correct pair is C and E? Wait, the problem says "two figures that have the same surface area but different volumes". Let's take Figure C (let's say it's 3x2x1, 6 cubes? Wait, no, the first figure (C) looks like 3 rows, 3 columns, 1 layer: 9 cubes. The second (D) is 6 cubes in a line. The third (E) has, let's count: bottom layer 2x3? No, looking at the colors, maybe E has 7 cubes. Wait, maybe the intended answer is C and D? No, surface area. Wait, maybe I made a mistake in calculation. Let's try again.

For a rectangular prism with \(l\), \(w\), \(h\):

• Figure C: Let's say \(l = 3\), \(w = 3\), \(h = 1\). Surface area \(S=2(lw + lh+wh)=2(9 + 3+3)=30\). Volume \(V = lwh=9\).

• Figure D: \(l = 6\), \(w = 1\), \(h = 1\). Surface area \(S=2(6\times1 + 6\times1+1\times1)=2(6 + 6 + 1)=26\). No.

• Figure E: Let's assume it's a 2x3x1 with an extra cube. Wait, maybe Figure C and Figure E. Let's say Figure E has dimensions that give surface area 30. Suppose Figure E has 7 cubes. Volume 7, surface area 30. Then C (volume 9, surface area 30) and E (volume 7, surface area 30) would be the pair.

Answer:

The two figures are C and E (assuming the figures are labeled C, D, E as in the image, with C having 9 unit cubes, E having 7 unit cubes, and both having a surface area of 30 square units).