Question

what is the volume of this figure?
2 ft.
4 ft. 2 ft.
6 ft. 4 ft.
3 ft.
6 ft.
blank cubic feet
submit

Explanation:

Step1: Calculate volume of large rectangle

The large rectangle (if there were no indentation) would have dimensions \( 6 \, \text{ft} \times 6 \, \text{ft} \times 3 \, \text{ft} \). Volume formula is \( V = l \times w \times h \), so \( V_{\text{large}} = 6 \times 6 \times 3 = 108 \, \text{cubic feet} \).

Step2: Calculate volume of the indentation

The indentation is a rectangular prism with dimensions \( 4 \, \text{ft} \times (6 - 2 - 2) \, \text{ft} \times 3 \, \text{ft} \). First, the width of the indentation: \( 6 - 2 - 2 = 2 \, \text{ft} \). Then volume: \( V_{\text{indent}} = 4 \times 2 \times 3 = 24 \, \text{cubic feet} \).

Step3: Subtract indentation from large volume

The volume of the figure is \( V_{\text{large}} - V_{\text{indent}} = 108 - 24 = 84 \, \text{cubic feet} \). Alternatively, we can calculate the volume by adding the volumes of the three rectangular prisms: left (\( 2 \times 6 \times 3 \)), middle (\( 2 \times (6 - 4) \times 3 \)), right (\( 2 \times 4 \times 3 \)). Left: \( 2 \times 6 \times 3 = 36 \), middle: \( 2 \times 2 \times 3 = 12 \), right: \( 2 \times 4 \times 3 = 24 \). Sum: \( 36 + 12 + 24 = 72 \)? Wait, no, earlier method was wrong. Wait, let's re - examine the figure.

Wait, correct approach: The figure can be divided into three parts. Left part: width 2 ft, height 6 ft, depth 3 ft: \( 2 \times 6 \times 3 = 36 \). Middle part: width (6 - 2 - 2)=2 ft, height (6 - 4)=2 ft, depth 3 ft: \( 2 \times 2 \times 3 = 12 \). Right part: width 2 ft, height 4 ft, depth 3 ft: \( 2 \times 4 \times 3 = 24 \). Sum: \( 36+12 + 24=72 \)? Wait, no, my first method was incorrect. Wait, let's look at the height. Wait, the total height is 6 ft, the right part has height 4 ft, the middle indentation: the vertical length of the indentation is 4 ft? Wait, no, the figure's height is 6 ft, the right block has height 4 ft, so the middle block (the indentation's vertical part) has height \( 6 - 4 = 2 \) ft. Wait, maybe better to use the first method correctly. The large outer dimensions: length 6, width 6, depth 3. But the indentation is in the middle, with height 4 ft? Wait, no, the depth is 3 ft (the front - back dimension). Let's re - analyze the figure.

Alternative approach: The figure is a 3D shape with length 6, depth 3, and we can think of the height as varying. The left column: width 2, height 6, depth 3: \( 2\times6\times3 = 36 \). The middle column: width (6 - 2 - 2)=2, height (6 - 4)=2, depth 3: \( 2\times2\times3 = 12 \). The right column: width 2, height 4, depth 3: \( 2\times4\times3 = 24 \). Sum: \( 36 + 12+24 = 72 \)? Wait, that can't be. Wait, maybe the depth is 3 ft, length 6 ft, and the height: left block height 6, middle block height (6 - 4)=2, right block height 4. Wait, no, the depth is 3, so each block's volume is width × height × depth.

Wait, another way: The overall dimensions if we consider the base as 6x3 (length x depth), and height varying. The total volume can be calculated as the area of the front face times depth. The front face area: area of the large rectangle (6x6) minus the area of the indentation (4x(6 - 2 - 2)) = 36 - 8 = 28? No, front face: left rectangle (2x6), middle rectangle (2x(6 - 4)), right rectangle (2x4). So front area: (2×6)+(2×2)+(2×4)=12 + 4+8 = 24. Then volume is front area × depth (3): 24×3 = 72? Wait, I think I made a mistake earlier. Let's check the dimensions again.

Looking at the figure: The total length (horizontal) is 6 ft, depth (into the page) is 3 ft. The left block: width 2 ft, height 6 ft, depth 3 ft: volume 2×6×3 = 36. The middle block: the indentation's vertical part? Wait, no, the indentation is a rectangular prism that is removed from the large block. Wait, the large block (if no indentation) would be 6 (length) × 3 (depth) × 6 (height). But there is a notch in the middle. The notch has height 4 ft, width (6 - 2 - 2)=2 ft, depth 3 ft. Wait, no, the height of the notch: the total height is 6 ft, and the right block has height 4 ft, so the notch's height is 4 ft? Wait, the figure's left side is 6 ft tall, the right side is 4 ft tall, and the notch is between them, with height 4 ft? I think my initial analysis was wrong. Let's use the correct method of dividing the figure into three rectangular prisms:

1. Left prism: width = 2 ft, height = 6 ft, depth = 3 ft. Volume: \( 2 \times 6 \times 3 = 36 \) cubic feet.

2. Middle prism: width = (6 - 2 - 2) = 2 ft, height = (6 - 4) = 2 ft, depth = 3 ft. Volume: \( 2 \times 2 \times 3 = 12 \) cubic feet.

3. Right prism: width = 2 ft, height = 4 ft, depth = 3 ft. Volume: \( 2 \times 4 \times 3 = 24 \) cubic feet.

Now, sum these volumes: \( 36 + 12 + 24 = 72 \) cubic feet. Wait, but let's check with another method. The large rectangular prism (without the notch) has dimensions 6 (length) × 3 (depth) × 6 (height), volume \( 6 \times 3 \times 6 = 108 \). The notch is a rectangular prism with dimensions: length = (6 - 2 - 2)=2 ft, height = 4 ft, depth = 3 ft. Volume of notch: \( 2 \times 4 \times 3 = 24 \). Then the volume of the figure is \( 108 - 24 = 84 \). Wait, now I'm confused. Let's look at the height of the notch. The left side is 6 ft, the right side is 4 ft, so the notch's height should be 4 ft? Or is the notch's height 6 - 4 = 2 ft?

Wait, the figure: the left block is 6 ft tall, the right block is 4 ft tall, and the middle block (the part that is indented) is between them. The depth is 3 ft. Let's calculate the front - face area:

- Left rectangle: width 2, height 6: area = 12.

- Middle rectangle: width (6 - 2 - 2)=2, height (6 - 4)=2: area = 4.

- Right rectangle: width 2, height 4: area = 8.

Total front - face area: 12 + 4+8 = 24. Then volume is front - face area × depth (3): 24×3 = 72. Wait, but why the discrepancy? Let's check the dimensions again. The length (horizontal) is 6 ft, which is divided into three parts: 2 ft (left), 2 ft (middle), 2 ft (right). Yes, 2 + 2 + 2 = 6. The height: left is 6 ft, middle is (6 - 4)=2 ft (since the right is 4 ft tall), right is 4 ft. Depth is 3 ft. So:

- Left volume: 2 (width) × 6 (height) × 3 (depth) = 36.

- Middle volume: 2 (width) × 2 (height) × 3 (depth) = 12.

- Right volume: 2 (width) × 4 (height) × 3 (depth) = 24.

Sum: 36 + 12 + 24 = 72. I think my initial mistake was in the large block calculation. The large block would be 6×3×6, but that's incorrect because the right block is only 4 ft tall, not 6 ft. So the correct way is to divide into three parts. So the volume is 72 cubic feet? Wait, no, let's use the formula for composite solids. Another way: the figure can be seen as a large rectangular prism with length 6, depth 3, height 6, minus a smaller rectangular prism with length 4, depth 3, width (6 - 2 - 2)=2. Wait, no, the smaller prism's dimensions: length (height) 4, width (6 - 2 - 2)=2, depth 3. So volume to subtract: 4×2×3 = 24. Large prism: 6×3×6 = 108. 108 - 24 = 84. Now I'm really confused. Let's look at the figure again. The left side is 6 ft, the right side is 4 ft, and the indentation is 4 ft tall (the vertical side of the indentation is 4 ft). The horizontal length of the indentation: 6 - 2 - 2 = 2 ft (since left block is 2 ft, right block is 2 ft, so middle is 2 ft). The depth is 3 ft. So the indentation is a rectangular prism with dimensions: height 4 ft, width 2 ft, depth 3 ft. The large prism (if no indentation) would have height 6 ft, width 6 ft, depth 3 ft. But the indentation is not the full height of 6 ft, only 4 ft? Wait, no, the left block is 6 ft tall, so the indentation's height should be 6 ft? No, the right block is 4 ft tall, so the indentation is between the left (6 ft) and right (4 ft) blocks, so the indentation's height is 4 ft? I think the correct way is to calculate the volume as the sum of the three blocks: left (2x6x3), middle (2x(6 - 4)x3), right (2x4x3). So left: 36, middle: 2x2x3 = 12, right: 24. Sum: 36 + 12 + 24 = 72. Or, the front face area: the left rectangle is 2x6, middle is 2x(6 - 4)=2x2, right is 2x4. So front area: 12 + 4 + 8 = 24. Volume: 24x3 = 72. Yes, that makes sense. So the correct volume is 72 cubic feet? Wait, no, let's check with actual numbers. If we have a block that is 2x6x3 (left), volume 36. A block that is 2x2x3 (middle), volume 12. A block that is 2x4x3 (right), volume 24. 36 + 12 is 48, 48 + 24 is 72. So the volume is 72 cubic feet. I think my initial mistake was in the large prism's height. The large prism can't be 6 ft tall for the entire width because the right block is only 4 ft tall. So the correct volume is 72.

Answer:

72