Question

( )
area of a =
area of b =
area of c =
area of d =
area of e =
area of f =
surface area =
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Explanation:

Step1: Identify the dimensions of the rectangular prism (or cube - like shape here, but it's a rectangular prism with length 10 in, width 8 in, height 12 in? Wait, the left figure has 8 in (width), 10 in (length), 12 in (height). Wait, actually, in a rectangular prism, the surface area is calculated by finding the area of each face. The net has 6 faces: A, B, C, D, E, F. Let's assume the dimensions: let's check the left figure: length \( l = 10 \) in, width \( w = 8 \) in, height \( h = 12 \) in.

Step2: Area of A (let's say A is a face with dimensions \( l \times w \)? Wait, no, maybe A is \( 8 \times 10 \)? Wait, no, let's look at the net. The middle square - like part D: maybe D is \( 10 \times 8 \)? Wait, maybe the dimensions are length 10, width 8, height 12. So the faces:

• Face A: Let's say A is \( 8 \times 12 \). Area \( A = 8\times12 = 96 \) in²? Wait, no, maybe I got the dimensions wrong. Wait the left figure: the base is 10 in (length), 8 in (width), and height 12 in. So the three pairs of faces:

1. Front/Back: \( 10 \times 12 \) each
2. Left/Right: \( 8 \times 12 \) each
3. Top/Bottom: \( 10 \times 8 \) each

Now, looking at the net:

• A: Let's say A is a face with \( 10 \times 8 \)? No, wait the net: D is in the center. Let's assume:

• A: \( 10 \times 8 \) (area \( 80 \) in²)
• B: \( 10 \times 12 \) (area \( 120 \) in²)
• C: \( 8 \times 12 \) (area \( 96 \) in²)
• D: \( 10 \times 8 \) (area \( 80 \) in²)
• E: \( 8 \times 12 \) (area \( 96 \) in²)
• F: \( 10 \times 12 \) (area \( 120 \) in²)

Wait, maybe I mixed up. Let's re - check. The left figure: length \( l = 10 \), width \( w = 8 \), height \( h = 12 \).

So:

• Area of A: If A is a face with \( l \times w = 10\times8 = 80 \) in²
• Area of B: \( l \times h = 10\times12 = 120 \) in²
• Area of C: \( w \times h = 8\times12 = 96 \) in²
• Area of D: \( l \times w = 10\times8 = 80 \) in² (since D is opposite to A)
• Area of E: \( w \times h = 8\times12 = 96 \) in² (opposite to C)
• Area of F: \( l \times h = 10\times12 = 120 \) in² (opposite to B)

Then Surface Area: Sum of all areas: \( 80 + 120 + 96 + 80 + 96 + 120 \)

Calculate that: \( (80\times2)+(120\times2)+(96\times2)=160 + 240+192 = 592 \) in². Wait, but let's do step by step for each area:

Step3: Area of A: Assume A is a face with dimensions \( 10 \times 8 \). So \( A = 10\times8 = 80 \) in².

Step4: Area of B: Assume B is \( 10 \times 12 \). So \( B = 10\times12 = 120 \) in².

Step5: Area of C: Assume C is \( 8 \times 12 \). So \( C = 8\times12 = 96 \) in².

Step6: Area of D: D is same as A, so \( D = 80 \) in².

Step7: Area of E: E is same as C, so \( E = 96 \) in².

Step8: Area of F: F is same as B, so \( F = 120 \) in².

Step9: Surface Area: Sum all areas: \( 80 + 120 + 96 + 80 + 96 + 120 = (80 + 80)+(120 + 120)+(96 + 96)=160 + 240 + 192 = 592 \) in².

Wait, but maybe the dimensions are different. Wait the left figure: the width is 8 in (the side), length 10 in (the base), height 12 in (the vertical side). So the three face areas:

• \( 10 \times 8 = 80 \)
• \( 10 \times 12 = 120 \)
• \( 8 \times 12 = 96 \)

Each of these areas appears twice in the net (since a rectangular prism has 6 faces: 2 of each type). So:

• Area of A: 80 (if A is \( 10 \times 8 \))
• Area of B: 120 (if B is \( 10 \times 12 \))
• Area of C: 96 (if C is \( 8 \times 12 \))
• Area of D: 80 (same as A)
• Area of E: 96 (same as C)
• Area of F: 120 (same as B)
• Surface Area: \( 2\times(80 + 120 + 96)=2\times296 = 592 \) in².

Answer:

• Area of A: \( \boldsymbol{80} \) in² (if A is \( 10\times8 \))
• Area of B: \( \boldsymbol{120} \) in² (if B is \( 10\times12 \))
• Area of C: \( \boldsymbol{96} \) in² (if C is \( 8\times12 \))
• Area of D: \( \boldsymbol{80} \) in² (same as A)
• Area of E: \( \boldsymbol{96} \) in² (same as C)
• Area of F: \( \boldsymbol{120} \) in² (same as B)
• Surface Area: \( \boldsymbol{592} \) in²

(Note: The exact values depend on the correct identification of the face dimensions. If the dimensions are different, the areas will change. For example, if the width is 8, length 10, height 12, the above calculations hold.)