Question

dara is building a plant stand. she wants to stain all sides of both pieces of the plant stand, except for the bottom of the larger prism. find the surface area of the part of the plant stand she will stain.
image of a composite prism: larger prism with side length 12 in (cube-like), smaller prism with side length 9 in attached to it
the surface area is

square inches.

Explanation:

Step1: Surface area of larger cube

The larger prism is a cube with side length \( s = 12 \) in. The surface area of a cube is \( 6s^2 \), but we need to adjust for the area where the smaller cube is attached. First, calculate the surface area of the larger cube without considering the attachment: \( 6\times12^2 = 6\times144 = 864 \) square inches. But we have a square area where the smaller cube is attached, which is \( 9\times9 \) (since the smaller cube has side 9 in), and this area is subtracted from the larger cube's surface area but added back as the surface area of the smaller cube's sides (except the one attached to the larger cube). Wait, actually, the correct approach is: the total surface area is the surface area of the larger cube plus the surface area of the smaller cube minus twice the area of the overlapping face (because the overlapping face is internal and not stained).

Step2: Surface area of smaller cube

The smaller cube has side length \( s = 9 \) in. Its surface area is \( 6\times9^2 = 6\times81 = 486 \) square inches. But the face where it's attached to the larger cube (area \( 9\times9 = 81 \)) is not stained, so we subtract that once from the smaller cube's surface area. Also, the larger cube had that face (area 81) covered, so we subtract that once from the larger cube's surface area. Wait, no: when we put the two together, the total surface area is (surface area of larger cube - area of one face) + (surface area of smaller cube - area of one face). Because the overlapping face is internal and not part of the stained area.

So, surface area of larger cube: \( 6\times12^2 - 9\times9 = 864 - 81 = 783 \)

Surface area of smaller cube: \( 6\times9^2 - 9\times9 = 486 - 81 = 405 \)

Then total surface area: \( 783 + 405 = 1188 \)? Wait, no, that's wrong. Wait, actually, the larger cube's surface area: originally, a cube has 6 faces. When we attach the smaller cube, we are covering one face of the larger cube with the smaller cube's face, so we lose that area from the larger cube, but we gain the other 5 faces of the smaller cube. Wait, yes! Because the bottom of the larger prism is not stained, but the problem says "all sides of both pieces of the plant stand, except for the bottom of the larger prism". Wait, the problem says: "stain all sides of both pieces of the plant stand, except for the bottom of the larger prism". Oh! I missed that. The bottom of the larger prism is not stained. So first, let's re-express:

Larger prism: it's a cube (12x12x12), but bottom is not stained. So surface area of larger prism: 5 faces (top, front, back, left, right) plus the area of the face where the smaller cube is attached? Wait, no, the larger prism has a part cut out? Wait, looking at the diagram: the larger prism is 12x12x12, and there's a smaller cube (9x9x9) attached? Wait, no, the diagram shows a larger cube (12x12x12) with a smaller cube (9x9x9) attached to one of its faces, but the bottom of the larger prism is not stained. Wait, maybe the larger prism is a cube with a smaller cube attached to its side, and the bottom of the larger prism (the base) is not stained.

Let's re-express:

- Larger prism: dimensions 12x12x12. Bottom face (12x12) is not stained. So surface area of larger prism: 5 faces (top, front, back, left, right) plus the area of the face where the smaller cube is attached (but wait, the smaller cube is attached to one of the side faces, not the bottom). Wait, the diagram: the larger cube has 12 in sides, and the smaller cube (9 in sides) is attached to one of its side faces. The bottom of the larger prism (the base) is 12x12, not stained.

So first, calculate the surface area of the larger prism excluding the bottom:

Larger prism: 5 faces (each 12x12) plus the area of the face where the smaller cube is attached, but then we have to consider the smaller cube's surface area (excluding the face attached to the larger prism) and also the area of the larger prism's face that's covered by the smaller cube? Wait, no. Let's think of the composite figure:

- The larger prism: bottom face (12x12) is not stained. The other 5 faces: top (12x12), front (12x12), back (12x12), left (12x12), right (12x12 - but wait, the right face has a part cut out? No, the diagram shows the larger cube with a smaller cube attached to its right face. Wait, the larger cube is 12x12x12, and the smaller cube is 9x9x9, attached to the right face of the larger cube. So the right face of the larger cube: originally 12x12, but now has a 9x9 square covered by the smaller cube, so the exposed area of the larger cube's right face is 12x12 - 9x9. Then the smaller cube has 5 faces exposed (since one face is attached to the larger cube). Also, the bottom of the larger prism is not stained (12x12), and the bottom of the smaller prism: is it stained? The problem says "all sides of both pieces of the plant stand, except for the bottom of the larger prism". So the bottom of the smaller prism: is that stained? Wait, the problem says "all sides of both pieces... except for the bottom of the larger prism". So sides: so bottom of larger is not stained, but bottom of smaller: is that a side? Wait, "sides" usually means lateral sides, not top or bottom? Wait, the problem says "all sides of both pieces of the plant stand, except for the bottom of the larger prism". So "sides" here probably means all faces except the bottom of the larger prism. So bottom of larger: not stained. Bottom of smaller: is that stained? Let's re-read: "stain all sides of both pieces of the plant stand, except for the bottom of the larger prism". So "sides" includes all faces except the bottom of the larger prism. So bottom of larger: no. Bottom of smaller: yes, because it's a side of the smaller piece, and the exception is only the bottom of the larger.

So let's re-calculate:

Larger prism:

- Bottom face: 12x12, not stained.

- Top face: 12x12, stained.

- Front, back, left faces: each 12x12, stained.

- Right face: 12x12, but with a 9x9 square covered by the smaller cube, so exposed area: 12x12 - 9x9 = 144 - 81 = 63. But also, the smaller cube's left face (attached to larger) is not stained, but the other faces of the smaller cube are stained.

Smaller prism (cube):

- Left face: attached to larger, not stained.

- Top, front, back, right, bottom faces: stained. So 5 faces, each 9x9.

So surface area of larger prism:

Top: 12x12 = 144

Front: 12x12 = 144

Back: 12x12 = 144

Left: 12x12 = 144

Right: 12x12 - 9x9 = 63

Bottom: 0 (not stained)

Wait, no, the larger prism is a 3D shape: length, width, height 12 in. So its surface area (excluding bottom) is:

Top: 12x12 = 144

Front: 12x12 = 144

Back: 12x12 = 144

Left: 12x12 = 144

Right: 12x12 = 144 (but with a 9x9 square covered, so we have to subtract that 9x9 and add the smaller cube's other faces)

Wait, I think the correct formula is:

Total surface area = (surface area of larger cube - bottom face - area of overlapping face) + (surface area of smaller cube - area of overlapping face)

Because:

- Larger cube: surface area is 6*12², but we exclude the bottom face (12²) and also the overlapping face (9²) because it's covered by the smaller cube.

- Smaller cube: surface area is 6*9², but we exclude the overlapping face (9²) because it's attached to the larger cube.

So let's calculate:

Larger cube: 6*12² - 12² - 9² = 5*12² - 9² = 5*144 - 81 = 720 - 81 = 639

Smaller cube: 6*9² - 9² = 5*9² = 5*81 = 405

Total surface area: 639 + 405 = 1044? Wait, no, that can't be. Wait, no: the larger cube's surface area excluding bottom is 5*12² (since bottom is excluded), and then we have the overlapping face: when we attach the smaller cube, we are covering a part of the larger cube's side face (not the bottom). So the larger cube's side face (one of the 4 side faces, not top or bottom) has a 9x9 area covered, so we subtract that area from the larger cube's surface area, and add the 5 faces of the smaller cube.

Wait, let's start over:

1. Larger prism (cube) dimensions: 12x12x12. Bottom face (12x12) is not stained. So surface area of larger prism (excluding bottom) is: top (12x12) + front (12x12) + back (12x12) + left (12x12) + right (12x12) = 5*12*12 = 5*144 = 720. But the right face has a 9x9 square covered by the smaller cube, so we subtract 9x9 from the right face's area: 144 - 81 = 63. So larger prism's surface area: 144 (top) + 144 (front) + 144 (back) + 144 (left) + 63 (right) = 144*4 + 63 = 576 + 63 = 639.

2. Smaller prism (cube) dimensions: 9x9x9. It is attached to the larger prism's right face, so its left face (9x9) is not stained. The other faces: top (9x9), front (9x9), back (9x9), right (9x9), bottom (9x9). So surface area of smaller prism: 5*9*9 = 5*81 = 405.

3. Total surface area: 639 + 405 = 1044. Wait, but let's check with another method.

Alternative method:

Surface area of larger cube (excluding bottom): 5*12² = 720.

Surface area of smaller cube (excluding the face attached to larger cube): 5*9² = 405.

But we also have to consider that the larger cube's right face has a 9x9 area covered, so we subtract that from the larger cube's surface area, and add the smaller cube's 5 faces. Wait, no, the larger cube's right face area is 12x12, but we subtract 9x9 (the covered area) and add the smaller cube's 5 faces. So:

Larger cube (excluding bottom): 5*12² = 720. But one of the side faces (right) has a 9x9 area covered, so we subtract 9x9: 720 - 81 = 639.

Smaller cube: 5*9² = 405.

Total: 639 + 405 = 1044.

Wait, but let's check with the formula for composite solids. The total surface area is the sum of the surface areas of the two solids minus twice the area of the overlapping face (because the overlapping face is internal and not part of the surface area of either solid).

Surface area of larger cube: 6*12² = 864. Exclude bottom: 864 - 12² = 864 - 144 = 720.

Surface area of smaller cube: 6*9² = 486.

Overlapping face area: 9² = 81.

Total surface area = (720 + 486) - 2*81 = 1206 - 162 = 1044. Yes! That matches. Because we have to subtract the overlapping face from both solids (since it's internal), so twice the area.

Yes, that's the correct formula: total surface area = (surface area of solid 1 - area of excluded face) + (surface area of solid 2) - 2*area of overlapping face.

Here, solid 1 (larger) has bottom face excluded: surface area of solid 1 (including bottom) is 6*12², so excluding bottom is 6*12² - 12² = 5*12² = 720.

Solid 2 (smaller) has surface area 6*9² = 486.

Overlapping face area: 9² = 81.

So total surface area = 720 + 486 - 2*81 = 1206 - 162 = 1044.

Answer:

1044