Question

1. the cylinder pictured below has a radius of 3 cm and a total surface area of 150 cm². the cube pictured below has a surface area of 216 cm². if the cylinder is placed on top of the cube and centered, what will the total surface area be? a 422.55 cm² b 366.00 cm² c 337.73 cm² d 309.45 cm² work space

Explanation:

Step1: Find the area of the circular base of the cylinder

The formula for the area of a circle is \( A = \pi r^2 \). Given \( r = 3 \) cm, we have \( A=\pi\times3^2 = 9\pi\) \( \text{cm}^2 \).

Step2: Analyze the total surface area of the combined figure

When the cylinder is placed on the cube, the overlapping area (the base of the cylinder and the corresponding area on the cube) is subtracted twice (once from the cylinder's surface area and once from the cube's, but actually we need to add the lateral surface area of the cylinder to the cube's surface area, and subtract the area of the two circular bases? Wait, no. Wait, the total surface area of the cylinder is \( 150 \) \( \text{cm}^2 \), which includes two circular bases and the lateral surface area. The cube's surface area is \( 216 \) \( \text{cm}^2 \). When we place the cylinder on the cube, we are covering one face of the cube with the base of the cylinder, so we need to subtract the area of one circular base from the cylinder's surface area (because that base is now attached to the cube and not exposed) and also subtract the area of one face of the cube (but wait, no: the cube's surface area is 6 faces. When we put the cylinder on top, we cover one face of the cube, so we need to subtract the area of that face? Wait, no, actually, the correct approach is: the combined surface area is (surface area of cylinder - area of one circular base) + (surface area of cube - area of one face of the cube) + area of the circular base? No, that's not right. Wait, let's think again. The cylinder has a total surface area of \( 2\pi r^2 + 2\pi r h=150 \) (where \( h \) is the height of the cylinder). The cube has a surface area of \( 6s^2 = 216 \), so \( s^2=36 \), so \( s = 6 \) cm (since side length \( s \) of cube is positive). Now, when we place the cylinder on the cube, the area where they are joined (the base of the cylinder and the top face of the cube) is no longer exposed. So for the cylinder, we now have the lateral surface area (total surface area of cylinder minus two circular bases) plus one circular base? Wait, no. Wait, the total surface area of the cylinder is \( 2\pi r^2 + 2\pi r h = 150 \). The lateral surface area is \( 2\pi r h=150 - 2\times9\pi=150 - 18\pi \). The cube's surface area is \( 216 \), which is \( 6\times6\times6 \) (since \( 6^3 \) is volume, wait no, surface area of cube is \( 6s^2 \), so \( 6s^2 = 216 \implies s^2 = 36 \implies s = 6 \) cm. So the top face of the cube has area \( 6\times6 = 36 \) \( \text{cm}^2 \). When we place the cylinder on the cube, we are covering the top face of the cube with the base of the cylinder. So the combined surface area is (surface area of cylinder - area of one circular base) + (surface area of cube - area of one face of the cube) + area of the circular base? No, that's confusing. Wait, actually, the correct way is: the combined surface area is (surface area of cylinder - area of one circular base) + surface area of cube. Because the cylinder's top base is still exposed, but the bottom base is attached to the cube, so we subtract one circular base from the cylinder's surface area (since that base is now internal), and the cube's surface area is reduced by the area of the face covered by the cylinder's base, but wait, no: the cube's surface area is 6 faces. When we put the cylinder on top, we cover one face, so we need to subtract the area of that face from the cube's surface area? Wait, no, the cube's surface area is 216, which is 6 faces each of area 36. If we cover one face with the cylinder's base (area \( 9\pi \approx 28.27 \) \( \text{cm}^2 \)), but wait, the area of the cube's face is 36, and the cylinder's base is 9π≈28.27. Wait, maybe my initial approach was wrong. Wait, let's re - express:

Total surface area of combined figure = (Total surface area of cylinder - area of one circular base) + (Total surface area of cube - area of one face of the cube) + area of the circular base? No, that cancels out. Wait, no. Let's think of the cylinder: its total surface area is 150, which includes two circular bases (each of area \( 9\pi \)) and the lateral surface area (\( 2\pi r h \)). When we place it on the cube, the bottom circular base of the cylinder is now attached to the cube, so we don't want to include that base in the total surface area. So the exposed surface area of the cylinder is \( 150 - 9\pi \). The cube's surface area is 216, but we have covered one face of the cube with the bottom base of the cylinder. However, the area of the cube's face is \( s^2 \), and the area of the cylinder's base is \( 9\pi \). Wait, but the cube's face area is \( 6\times6 = 36 \) (since \( 6s^2=216\implies s = 6 \)). So the area of the cube's face that is covered is 36, but the cylinder's base covers only \( 9\pi\approx28.27 \) of it? No, that can't be. Wait, no, the cylinder is centered on the cube, so the base of the cylinder must fit on the face of the cube. Since the side length of the cube is 6 cm, and the radius of the cylinder is 3 cm, the diameter of the cylinder is 6 cm, which is equal to the side length of the cube. So the base of the cylinder (a circle with diameter 6 cm) exactly fits on the face of the cube (a square with side length 6 cm). So the area of the base of the cylinder is equal to the area of the part of the cube's face that is covered. So the area of the base of the cylinder is \( \pi r^2=\pi\times3^2 = 9\pi\approx28.27 \), but the area of the cube's face is \( 6\times6 = 36 \). Wait, no, if the diameter of the cylinder is 6 cm (since radius 3 cm), then the base of the cylinder is a circle with diameter 6 cm, which has the same width as the cube's face (side length 6 cm), so the circle is inscribed in the square. So the area of the base of the cylinder is \( 9\pi\approx28.27 \), and the area of the cube's face is 36. But when we place the cylinder on the cube, the exposed surface area of the cube is its total surface area minus the area of the face covered by the cylinder, and the exposed surface area of the cylinder is its total surface area minus the area of the base that is attached to the cube. Then we add the lateral surface area of the cylinder? No, the cylinder's total surface area includes the two bases and the lateral surface area. So when we place the cylinder on the cube, we need to:

1. Take the cube's surface area: 216.
2. Take the cylinder's surface area: 150.
3. Subtract twice the area of the base of the cylinder? No, wait, no. Wait, the cylinder has two bases (top and bottom) and lateral surface area. The bottom base is attached to the cube, so we subtract the area of the bottom base from the cylinder's surface area (because that base is no longer exposed). The cube's top face has an area of 36, but the part covered by the cylinder's bottom base is \( 9\pi \), but since the cylinder is centered, the area of the cube's top face that is exposed is \( 36 - 9\pi \), and the cylinder's bottom base is not exposed. Also, the cylinder's top base is exposed, and its lateral surface area is exposed. Wait, I think I made a mistake earlier. Let's start over.

The total surface area of the combined figure is:

Surface area of cube + surface area of cylinder - 2×(area of the base of the cylinder)

Wait, why? Because when we put the cylinder on the cube, we are covering one base of the cylinder (so we subtract it from the cylinder's surface area) and covering one face of the cube (but the face of the cube is part of its surface area, so we subtract the area of the base of the cylinder from the cube's surface area as well? No, no. The correct formula is: when two solids are joined, the total surface area is the sum of their individual surface areas minus twice the area of the overlapping region (because the overlapping region is internal and not exposed on either solid). In this case, the overlapping region is the area where the cylinder's base meets the cube's face. The area of the overlapping region is the area of the base of the cylinder (since the cylinder's base is on the cube's face). So we need to subtract 2×(area of the base of the cylinder) from the sum of the surface areas of the cylinder and the cube.

So:

Total surface area = (Surface area of cube) + (Surface area of cylinder) - 2×(Area of base of cylinder)

We know:

- Surface area of cube = 216 \( \text{cm}^2 \)
- Surface area of cylinder = 150 \( \text{cm}^2 \)
- Area of base of cylinder = \( \pi r^2=9\pi \) \( \text{cm}^2 \)

So:

Total surface area = \( 216 + 150-2\times9\pi \)

Calculate \( 2\times9\pi = 18\pi\approx18\times3.1416 = 56.5488 \)

Then \( 216 + 150=366 \), and \( 366-56.5488 = 309.4512\approx309.45 \) \( \text{cm}^2 \)

Answer:

D. \( 309.45\ \text{cm}^2 \)