Question

a briefcase will be covered in a vinyl and leather combination. the leather covers the two largest sides and vinyl covers the smaller sides. if the briefcase measures 1 foot 8 inches by 12 inch and is 4 inches deep, what is the total area that must be covered? briefcase front back 1 ft 1 ft 1 ft 8 in 1 ft 8 in side side 1 ft 1 ft top bottom 4 in 4 in 1 ft 8 in 1 ft 8 in

Explanation:

Step1: Convert units to inches

1 foot 8 inches = 12 + 8 = 20 inches. So the dimensions are 20 in (length) × 12 in (width) × 4 in (depth).

Step2: Identify largest and smaller sides

Largest sides: front/back (area = length × height) and top/bottom (area = length × width)? Wait, no—wait, the briefcase: front/back (20×12), top/bottom (20×4), sides (12×4). Wait, no, let's recalculate areas:

• Front/Back: 20 in × 12 in = 240 in² each. Two of them: 2×240 = 480 in² (leather).
• Top/Bottom: 20 in × 4 in = 80 in² each. Two of them: 2×80 = 160 in² (vinyl? Wait, no, the problem says "leather covers the two largest sides and vinyl covers the smaller sides". Wait, maybe I misidentified. Wait, let's list all faces:
• Front/Back: 20×12 = 240
• Top/Bottom: 20×4 = 80
• Left/Right (sides): 12×4 = 48

So largest area per face: 240 (front/back), then 80 (top/bottom), then 48 (sides). Wait, but the problem says "two largest sides"—so front and back (two sides, each 240) are leather. Then the other sides: top, bottom, left, right? Wait, no, a briefcase has 6 faces: front, back, top, bottom, left, right. Wait, the diagram shows front, back, two sides (left/right), top, bottom. So:

• Leather: two largest sides. Let's calculate each face area:
• Front/Back: 20×12 = 240 (each)
• Top/Bottom: 20×4 = 80 (each)
• Sides (left/right): 12×4 = 48 (each)

So the two largest faces are front and back (240 each). Then the smaller sides: top, bottom, left, right? Wait, no, the problem says "vinyl covers the smaller sides"—so how many smaller sides? Wait, maybe the briefcase is a rectangular prism. So total surface area is 2(lw + lh + wh). But here, leather is two largest faces, vinyl is the rest. Wait, let's re-express:

Wait, the briefcase dimensions: length (l) = 20 in, width (w) = 12 in, height (h) = 4 in.

Faces:

• l×w: 20×12 = 240 (front/back: 2 faces)
• l×h: 20×4 = 80 (top/bottom: 2 faces)
• w×h: 12×4 = 48 (left/right: 2 faces)

So the two largest faces (by area) are the l×w faces (240 each). So leather covers these two: 2×240 = 480.

Vinyl covers the remaining faces: l×h (two faces) and w×h (two faces). Wait, but the problem says "vinyl covers the smaller sides"—so maybe "smaller sides" are the ones with smaller area. The l×h faces (80) and w×h (48). Wait, but the problem says "the smaller sides"—maybe "sides" here refer to the four other sides (top, bottom, left, right). Wait, let's check the diagram: the diagram shows front, back (1 ft 8 in × 1 ft? Wait, no, the diagram labels front/back as 1 ft 8 in (20 in) × 1 ft (12 in), top/bottom as 1 ft 8 in × 4 in, sides as 1 ft × 4 in. So:

Front/Back: 20 in × 12 in (leather, two sides)
Top/Bottom: 20 in × 4 in (vinyl, two sides)
Sides (left/right): 12 in × 4 in (vinyl, two sides)

So total area: leather (front + back) + vinyl (top + bottom + left + right)

Leather area: 2 × (20 × 12) = 2 × 240 = 480 in²

Vinyl area: 2 × (20 × 4) + 2 × (12 × 4) = 2×80 + 2×48 = 160 + 96 = 256 in²

Total area: 480 + 256 = 736 in²

Wait, let's verify:

Front/Back: 20*12=240 each, two: 480

Top: 204=80, Bottom: 80, Left:124=48, Right:48

So vinyl: 80+80+48+48=256

Total: 480+256=736.

Step3: Sum leather and vinyl areas

Leather: 2(2012) = 480

Vinyl: 2(204) + 2(124) = 160 + 96 = 256

Total: 480 + 256 = 736 in²

Answer:

736 square inches