d. 51
short answer
33. topic: sa
this object is composed of two identical cylinders connected by a right rectangular prism.
each cylinder has diameter 18 cm and height 4 cm.
the rectangular prism has length 8 cm and square ends of side length 2 cm.
determine the surface area of the object. give your answer to the nearest whole number.
(image of the object: two cylinders connected by a rectangular prism, with dimensions labeled: 4 cm, 18 cm, 2 cm, 8 cm)

Question

Accepted Answer

two cylinders are connected to the prism, each cylinder has one circular face (the outer one) exposed, one circular face (the inner one) with a square (2x2) cut out, and the lateral surface.

Wait, another approach: The total surface area of the composite object is:

- Lateral surface area of two cylinders.
- Area of two outer circular faces of the cylinders.
- Lateral surface area of the rectangular prism.
- Area of the two inner circular faces of the cylinders minus the area of the two square faces (where the prism is attached).

Wait, no, the inner circular face of each cylinder has an area of $ \pi r^2$, but a square of $ 2\times2$ is covered by the prism, so the exposed area of the inner circular face is $ \pi r^2 - 2^2$. And the outer circular face is fully exposed ($ \pi r^2$). The lateral surface area of each cylinder is $ 2\pi rh$.

So for one cylinder:

- Outer circular face: $ \pi r^2$
- Inner circular face: $ \pi r^2-2^2$
- Lateral surface area: $ 2\pi rh$

For two cylinders:

$ 2\times(\pi r^2+(\pi r^2 - 4)+2\pi rh)=2\times(2\pi r^2+2\pi rh - 4)=4\pi r^2+4\pi rh - 8$

For the prism:

- Lateral surface area: $ 2\times(8\times2 + 8\times2)=64$ (as the two square faces are attached to the cylinders, so we don't include them)

Now, $ r = 9$, $ h = 4$

$ 4\pi r^2=4\pi\times81 = 324\pi$

$ 4\pi rh=4\pi\times9\times4 = 144\pi$

So $ 324\pi+144\pi-8 + 64=468\pi+56$

$ 468\pi\approx468\times3.1416 = 1470.0688$

$ 1470.0688+56 = 1526.0688\approx1526$

Wait, but let's check with another method. Maybe the height of the cylinder is 18 cm? No, the diameter is 18 cm. Wait, maybe the "height" of the cylinder is 4 cm, and the length along the axis is 18 cm? No, the diagram shows the cylinder with diameter 18 cm and 4 cm as the height.

Wait, maybe I made a mistake in the cylinder's dimensions. Let's check the problem statement again: "Each cylinder has diameter 18 cm and height 4 cm. The rectangular prism has length 8 cm and square ends of side length 2 cm."

Ah! Wait a minute, the height of the cylinder is 4 cm, which is the length along the axis of the cylinder (i.e., the distance from one circular face to the other). So the radius $ r=\frac{18}{2}=9$ cm, height $ h = 4$ cm.

The lateral surface area of a cylinder is $ 2\pi rh=2\pi\times9\times4 = 72\pi$ per cylinder, so for two cylinders, $ 2\times72\pi = 144\pi$

The area of the two outer circular faces: $ 2\times\pi r^2=2\times\pi\times81 = 162\pi$

The area of the two inner circular faces: $ 2\times\pi r^2=2\times81\pi = 162\pi$, but we have to subtract the area of the two square faces (where the prism is attached) which is $ 2\times(2\times2)=8$

The lateral surface area of the prism: the prism has length 8 cm, and the perimeter of the square base is $ 4\times2 = 8$ cm, so lateral surface area is $ 8\times8 = 64$ (since lateral surface area of a prism is perimeter of base times height, where height is the length of the prism)

Now, total surface area:

Lateral surface area of two cylinders + area of two outer circular faces + area of two inner circular faces - area of two square faces + lateral surface area of prism

$$
\begin{align*}
SA&=144\pi+162\pi+162\pi - 8+64\
&=(144 + 162+162)\pi+56\
&=468\pi+56
\end{align*}
$$

Which is the same as before. Now, $ 468\pi\approx468\times3.1416 = 1470.0688$, $ 1470.0688+56 = 1526.0688\approx1526$

Wait, but let's verify with approximate values:

$ \pi\approx3.14$

$ 468\times3.14 = 468\times3+468\times0.14=1404 + 65.52 = 1469.52$

$ 1469.52+56 = 1525.52\approx1526$

Yes, so the surface area is approximately 1526 square centimeters.

# Answer:
1526

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