Question

find the volume of the object by counting the cubic units. assume that there are blocks you cannot see if they are necessary for the structure. find the surface area of the object by counting square units. the volume is \\(\square\\) cubic units. (simplify your answer.)

Explanation:

Step1: Analyze the structure (front - back, left - right, top - bottom layers)

Looking at the figure, we can split it into two parts: the taller part and the shorter part. Let's assume the side length of each small cube is 1 unit.
For the taller part (let's say it's a rectangular prism with dimensions \(4\times2\times5\)? Wait, no, better to count layers. Wait, actually, let's see the number of cubes in each layer. Wait, maybe a better way: the left part (taller) has a base of \(4\times2\) (width 4, depth 2) and height 5? No, wait the figure: looking at the front view, the left part is 4 columns (width) and 2 rows (depth) and height 5? Wait no, the right part is shorter, height 2. Wait, let's count the number of cubic units (volume) by counting the number of small cubes.

Let's break the object into two rectangular prisms:

1. The first prism (taller one): Let's say it has dimensions: length (along the depth) = 2, width (along the front - back) = 4, height = 5? Wait no, looking at the figure, the taller part is 4 units wide (horizontal), 2 units deep (into the page), and 5 units tall? No, wait the shorter part is 4 units wide (horizontal), 2 units deep, and 2 units tall? Wait, no, let's count the number of cubes:

Wait, the left (taller) section: in each layer (height), the number of cubes is \(4\times2\) (since width 4, depth 2). How many layers? 5? Wait no, the right (shorter) section: width 4, depth 2, height 2. Wait, no, maybe I got it wrong. Wait, looking at the figure, the taller part is 4 columns (x - direction) and 2 rows (y - direction) and 5 layers (z - direction)? No, the shorter part is attached to the taller part. Wait, actually, let's count the volume:

First, the taller rectangular prism: let's say its dimensions are length \(l_1 = 2\) (depth), width \(w_1 = 4\) (width), height \(h_1 = 5\)? No, that can't be. Wait, no, the shorter part: length \(l_2 = 2\), width \(w_2 = 4\), height \(h_2 = 2\). Wait, no, maybe the taller part is 4 (width) x 2 (depth) x 5 (height)? No, that would be \(4\times2\times5 = 40\). The shorter part is 4 (width) x 2 (depth) x 2 (height)? No, that would overlap. Wait, no, actually, the correct way is:

Wait, the object is composed of two parts:

- Part 1: A rectangular prism with dimensions: width = 4, depth = 2, height = 5. Wait, no, that's too big. Wait, no, looking at the figure, the taller part is 4 units wide (horizontal), 2 units deep (into the page), and 5 units tall? No, the shorter part is 4 units wide, 2 units deep, and 2 units tall, and it's attached to the taller part. Wait, no, maybe the taller part is 4 (width) x 2 (depth) x 5, and the shorter part is 4 (width) x 2 (depth) x 2, but they share a face, so we don't double - count. Wait, no, actually, the taller part is 4 (width) x 2 (depth) x 5, and the shorter part is 4 (width) x 2 (depth) x 2, but attached to the taller part, so the total volume is \(4\times2\times5+4\times2\times2\)? Wait, no, that would be if they are separate, but actually, the shorter part is attached to the taller part, so maybe the taller part is 4 (width) x 2 (depth) x 5, and the shorter part is 4 (width) x 2 (depth) x 2, but the overlapping area (where they are attached) is \(4\times2\times2\), so total volume is \(4\times2\times5 + 4\times2\times2\)? Wait, no, that's not right. Wait, no, actually, the taller part has height 5, and the shorter part has height 2, and they are attached along the height. Wait, maybe the correct dimensions:

Wait, let's count the number of cubes in each layer (z - direction, height):

- For height from z = 1 to z = 2 (lower 2 layers): the number of cubes is \((4 + 4)\times2\)? No, wait, no. Wait, the base (z = 1) and z = 2: the total width is \(4 + 4 = 8\)? No, the figure: the taller part is 4 units wide (left), and the shorter part is 4 units wide (right), but depth is 2 for both. Wait, no, depth is 2 (into the page) for both. So in the lower 2 layers (height 1 and 2), the number of cubes is \((4 + 4)\times2\) (width 8, depth 2)? No, that can't be. Wait, I think I made a mistake. Let's look at the figure again: the taller part is 4 columns (width) and 2 rows (depth) and 5 layers (height). The shorter part is 4 columns (width) and 2 rows (depth) and 2 layers (height), attached to the taller part at the bottom 2 layers. Wait, no, the taller part is 5 layers tall, and the shorter part is 2 layers tall, attached to the side of the taller part? No, the figure shows that the shorter part is attached to the taller part's lower part. Wait, maybe the correct way is:

The volume is the number of small cubic units. Let's count:

- The taller rectangular prism: length (depth) = 2, width (width) = 4, height = 5. So volume \(V_1 = 2\times4\times5=40\).

- The shorter rectangular prism: length = 2, width = 4, height = 2. So volume \(V_2 = 2\times4\times2 = 16\).

Wait, but do they overlap? No, because the shorter one is attached to the taller one, so total volume \(V = V_1+V_2=40 + 16=56\)? Wait, no, that can't be. Wait, maybe the taller part is 4 (width) x 2 (depth) x 5, and the shorter part is 4 (width) x 2 (depth) x 2, but actually, the width of the taller part is 4, and the shorter part is also width 4, but attached to the same depth? No, maybe the depth is 2 for both, so the total width is 4 (taller) + 4 (shorter) = 8? No, that would be width 8, depth 2, and height: the taller part is height 5, the shorter part is height 2. But that would mean in the lower 2 layers (height 1 and 2), the width is 8, depth 2, and in the upper 3 layers (height 3 - 5), the width is 4, depth 2.

Ah! That's the correct way. So:

- For height layers 1 and 2 (lower 2 layers): number of cubes per layer = \(8\times2 = 16\) (width 8, depth 2). So 2 layers: \(16\times2 = 32\).

- For height layers 3, 4, 5 (upper 3 layers): number of cubes per layer = \(4\times2 = 8\) (width 4, depth 2). So 3 layers: \(8\times3 = 24\).

Total volume \(V=32 + 24=56\). Wait, but let's check:

Lower 2 layers (height 1 and 2): width 8 (4 + 4), depth 2, so each layer has \(8\times2 = 16\) cubes. 2 layers: \(16\times2 = 32\).

Upper 3 layers (height 3 - 5): width 4, depth 2, each layer has \(4\times2 = 8\) cubes. 3 layers: \(8\times3 = 24\).

Total volume: \(32 + 24 = 56\).

Wait, but let's confirm with another approach. Let's count the number of cubes:

In the x - direction (width): the total width is 8 (4 from the taller part, 4 from the shorter part) in the lower 2 layers, and 4 in the upper 3 layers.

In the y - direction (depth): 2 (constant).

In the z - direction (height): 5 (total height).

So volume = (number of cubes in lower 2 layers) + (number of cubes in upper 3 layers)

= \((8\times2\times2)+(4\times2\times3)\)

= \((32)+(24)\)

= 56.

Yes, that makes sense.

Step2: Confirm the count

Wait, let's check the dimensions again. If the shorter part is attached to the taller part, in the lower 2 layers (height 1 and 2), the width is 4 (taller) + 4 (shorter) = 8, depth 2, so each layer has 8*2 = 16 cubes. 2 layers: 32. In the upper 3 layers (height 3,4,5), only the taller part exists, width 4, depth 2, so each layer has 4*2 = 8 cubes. 3 layers: 24. Total: 32 + 24 = 56.

Answer:

56