Question

select the correct answer from each drop-down menu. a composite figure is shown. (diagram includes labels: 6 ft, 6 ft, 6 ft, 20 ft, 4 ft) what is the surface area for each part of the figure? what is the total surface area of the figure? the surface area of the pyramid is dropdown square feet. the surface area of the square prism is dropdown square feet. the surface area of the cube is dropdown square feet. the total surface area is dropdown feet. (options like 60, 96 are visible.)

Explanation:

Step1: Surface Area of Pyramid

The pyramid has a square base with side 6 ft and slant height 4 ft. The surface area of a square pyramid (lateral + base, but here base is attached to prism, so lateral surface area: \(4\times\frac{1}{2}\times6\times4\))? Wait, no, wait. Wait, the base of the pyramid is a square with side 6 ft, and the triangular faces: each triangle has base 6, height (slant height) 4? Wait, no, maybe the height of the triangle is 4? Wait, the pyramid's lateral surface area: 4 triangles, each with area \(\frac{1}{2}\times6\times4\). So \(4\times\frac{1}{2}\times6\times4 = 48\)? Wait, no, wait, maybe the slant height is different. Wait, the cube is 6x6x6? Wait, the cube: surface area of a cube is \(6\times side^2\), but if one face is attached to the prism, then it's \(5\times6^2 = 180\)? Wait, no, let's re-examine.

Wait, the cube: each side is 6 ft. If it's a cube, surface area is \(6\times6\times6\)? No, surface area of a cube is \(6s^2\), where \(s = 6\), so \(6\times36 = 216\), but if one face is attached to the prism, then it's \(5\times36 = 180\). Wait, maybe the cube is 6x6x6, but one face is glued to the prism, so surface area of cube: \(5\times6^2 = 180\).

Square prism: length 20 ft, width 6 ft, height 6 ft? Wait, the prism has length 20, width 6, height 6. Surface area of a rectangular prism is \(2(lw + lh + wh)\), but if one face is attached to the cube and one face to the pyramid, so we need to subtract the overlapping areas. Wait, maybe the prism is a rectangular prism with length 20, width 6, height 6. But let's check the pyramid first.

Pyramid: square base (6x6), and four triangular faces. Each triangular face has base 6, and the slant height (height of the triangle) is 4? Wait, the diagram shows a 4 ft height for the pyramid's triangle? Wait, the lateral surface area of the pyramid (since the base is attached to the prism, so we don't include the base) is \(4\times\frac{1}{2}\times6\times4 = 48\)? Wait, no, maybe the slant height is different. Wait, maybe the pyramid's triangular faces have base 6 and height (slant height) calculated? Wait, no, the diagram shows 4 ft as the height from the center to the apex? Wait, no, the 4 ft is the height of the triangle? Wait, maybe I misread.

Wait, let's start over.

1. Cube: side length 6 ft. If it's a cube, and one face is attached to the prism, so surface area of cube: \(5\times6^2 = 5\times36 = 180\) square feet.

2. Square Prism: length 20 ft, width 6 ft, height 6 ft. But one face is attached to the cube (6x6) and one face to the pyramid (6x6), so we need to calculate its surface area minus the two attached faces? Wait, no, the prism's surface area: original surface area of rectangular prism is \(2(lw + lh + wh) = 2(20\times6 + 20\times6 + 6\times6) = 2(120 + 120 + 36) = 2(276) = 552\). But then we subtract the two faces that are attached (one to cube, one to pyramid), each 6x6, so 552 - 2*36 = 552 - 72 = 480? Wait, that can't be right. Wait, maybe the prism is a square prism with length 20, and the cross-section is 6x6. So the lateral faces: two faces of 20x6, and two faces of 20x6, and the two bases (6x6), but one base is attached to cube, one to pyramid. So lateral surface area (excluding the two bases) is \(2\times20\times6 + 2\times20\times6 = 480\)? No, wait, the cross-section is 6x6, so the perimeter of the base is 4*6=24, and lateral surface area is perimeter*length = 24*20=480. Then the two bases (6x6) are attached, so we don't include them. So surface area of prism is 480.

3. Pyramid: square base (6x6), but base is attached to prism, so we calculate lateral surface area. The pyramid has four triangular faces, each with base 6, and slant height. Wait, the diagram shows a 4 ft height from the center to the apex? Wait, no, the 4 ft is the height of the triangle? Wait, maybe the slant height is 5? Wait, no, the diagram has a 4 ft and 3 ft? Wait, maybe it's a right triangle with legs 3 and 4, hypotenuse 5. Oh! Wait, the pyramid's triangular face: the base is 6, so half of the base is 3, and the height of the triangle (slant height) is 5? Wait, 3-4-5 triangle. So the slant height is 5? Wait, the diagram shows 4 ft and 3 ft, so the slant height (height of the triangular face) is 5? Wait, let's check: if the base of the triangle is 6, then half of that is 3, and the height of the triangle (from the apex to the base) is 5 (since 3-4-5 triangle: 3, 4, 5). So the area of one triangular face is \(\frac{1}{2}\times6\times5 = 15\), so four faces: 4*15=60. Ah, that makes sense. So the surface area of the pyramid (lateral) is 60 square feet.

Now, cube: surface area. If it's a cube with side 6, and one face is attached to the prism, so we have 5 faces? Wait, no, the cube is attached to the prism, so the cube has a face (6x6) attached to the prism, so the cube's surface area is 6*6*6 - 6*6 = 216 - 36 = 180? Wait, no, a cube has 6 faces, each 6x6. If one face is glued to the prism, then we have 5 faces exposed? No, wait, the cube is part of the composite figure, so the surface area of the cube is the area of its faces not attached. Wait, the cube is 6x6x6, and one face is attached to the prism (6x6), so the surface area of the cube is 5*6*6 = 180? Wait, no, maybe the cube is actually a square prism with side 6 and length 6? Wait, no, the cube is 6x6x6.

Wait, let's re-express:

- Pyramid: square base (6x6), lateral surface area (4 triangles). Each triangle has base 6, height (slant height) 5 (since 3-4-5 triangle: half of base is 3, height of pyramid is 4, so slant height is 5). So area per triangle: 0.5*6*5=15, four triangles: 4*15=60. So surface area of pyramid is 60.

- Square Prism: length 20, width 6, height 6. Lateral surface area (excluding the two bases, which are attached to cube and pyramid) is 2*(20*6 + 20*6) = 2*(120 + 120) = 480? Wait, no, the prism has two rectangular faces of 20x6 (front and back) and two of 20x6 (left and right)? No, wait, the cross-section is 6x6, so the perimeter is 4*6=24, lateral surface area is perimeter*length=24*20=480. Then the two bases (6x6) are attached, so we don't include them. So surface area of prism is 480.

- Cube: side 6, and one face is attached to the prism (6x6), so the cube's surface area is 5*6*6=180? Wait, no, the cube is 6x6x6, so total surface area is 6*36=216, but one face (36) is attached to the prism, so we subtract that: 216 - 36=180.

Now total surface area: pyramid (60) + prism (480) + cube (180) - overlapping areas? Wait, no, the cube is attached to the prism (one face), and the prism is attached to the pyramid (one face). So the overlapping areas are already subtracted in each part. So total surface area is 60 + 480 + 180 = 720? Wait, but that seems high. Wait, maybe I made a mistake.

Wait, let's check the cube again. The cube: if it's a 6x6x6 cube, and one face is attached to the prism, then the surface area of the cube is 5*6*6=180. The prism: length 20, width 6, height 6. The surface area of the prism: the two ends (6x6) are attached to cube and pyramid, so we calculate the lateral surface area (the sides) which is 2*(20*6 + 20*6)=2*(120+120)=480. The pyramid: lateral surface area is 60 (as calculated with slant height 5). Then total surface area: 180 (cube) + 480 (prism) + 60 (pyramid) = 720.

But let's verify the pyramid's surface area again. The base of the pyramid is 6x6, but it's attached to the prism, so we don't include the base. The four triangular faces: each has a base of 6, and the slant height (height of the triangle) is 5 (since the pyramid's height is 4, and half the base is 3, so slant height is 5 by Pythagoras: 3² + 4² = 5²). So area of one triangle: 0.5*6*5=15, four triangles: 4*15=60. That's correct.

Cube: 5 faces, each 6x6: 5*36=180. Correct.

Prism: lateral surface area (since the two bases are attached) is perimeter of base * length. Base is 6x6, perimeter 24, length 20: 24*20=480. Correct.

Total: 180 + 480 + 60 = 720.

But let's check the drop-down options. The pyramid's surface area is 60 (as per the diagram's 60 option). The cube's surface area: 5*6*6=180? Wait, no, maybe the cube is a 6x6x6, but the surface area is 6*6*6? No, surface area of a cube is 6s², so 6*36=216, but if one face is attached, it's 216 - 36=180. The prism: maybe the prism is 20x6x6, and its surface area is 2*(20*6 + 20*6 + 6*6) - 2*36 (subtracting the two attached faces) = 2*(120+120+36) -72= 552 -72=480. Then total is 60+480+180=720.

So:

- Pyramid: 60

- Square Prism: 480

- Cube: 180

- Total: 60+480+180=720

But let's confirm the cube's surface area. If the cube is 6x6x6, and one face is attached to the prism, then the exposed surface area is 5 faces: 5*6*6=180. Correct.

The prism: length 20, width 6, height 6. The surface area without the two bases (6x6) is 2*(20*6 + 20*6)=480. Correct.

Pyramid: 60. Correct.

Total: 60+480+180=720.

Answer:

The surface area of the pyramid is \(\boldsymbol{60}\) square feet.
The surface area of the square prism is \(\boldsymbol{480}\) square feet.
The surface area of the cube is \(\boldsymbol{180}\) square feet.
The total surface area is \(\boldsymbol{720}\) square feet.