Question

what is the volume of this figure? figure with dimensions: 5 ft, 10 ft, 15 ft, 5 ft, 10 ft, 5 ft options: 1,000 cubic feet; 875 cubic feet; 750 cubic feet; 1,125 cubic feet

Explanation:

Step1: Divide the figure into three rectangular prisms

We can split the composite figure into three parts: left, middle, and right.
- Left prism: dimensions \(5\,\text{ft} \times 5\,\text{ft} \times 10\,\text{ft}\)
- Middle prism: dimensions \(15\,\text{ft} \times 5\,\text{ft} \times (10 - 5)\,\text{ft}=15\,\text{ft} \times 5\,\text{ft} \times 5\,\text{ft}\) (since the height of the middle part is the difference between the left height and the indentation)
- Right prism: dimensions \(5\,\text{ft} \times 5\,\text{ft} \times 15\,\text{ft}\)

Step2: Calculate volume of left prism

Volume of a rectangular prism is \(V = l \times w \times h\). For left prism:
\(V_1 = 5 \times 5 \times 10 = 250\) cubic feet.

Step3: Calculate volume of middle prism

For middle prism:
\(V_2 = 15 \times 5 \times 5 = 375\) cubic feet.

Step4: Calculate volume of right prism

For right prism:
\(V_3 = 5 \times 5 \times 15 = 375\) cubic feet.

Step5: Sum the volumes

Total volume \(V = V_1 + V_2 + V_3 = 250 + 375 + 375 = 1000\)? Wait, no, wait. Wait, maybe my division is wrong. Let's re - examine the figure.

Alternative approach: The figure can be seen as a large rectangular prism minus a smaller one? Wait, no. Wait, the correct way: Let's look at the base. The total length: \(5 + 15+ 5=25\) ft? No, wait the figure: left part is 5ft (width) x5ft (depth) x10ft (height). Middle part: 15ft (length) x5ft (depth) x5ft (height, since the right part is 15ft height, left is 10ft, so middle is 10 - 5? No, maybe better to split into two parts: left - middle combined and right.

Wait, another way: The right part is \(5\times5\times15 = 375\). The left - middle part: length is \(5 + 15=20\) ft, width 5ft, height 10ft? No, no. Wait the figure: the bottom part: from left, 5ft (width) x5ft (depth) x10ft (height). Then, above the bottom part, in the middle, there is a part? No, the figure is like:

Wait, the given dimensions: left part: 5ft (width) x5ft (depth) x10ft (height). Then, a middle part that is 15ft (length) x5ft (depth) x5ft (height) (since the right part is 15ft tall, left is 10ft, so the middle part's height is 15 - 10 = 5? No, I think I messed up.

Wait, let's use the correct method: The figure can be divided into three rectangular prisms:

1. Bottom left: \(5\times5\times10\)
2. Middle: \(15\times5\times5\) (because the height from the bottom to the top of the middle is 5ft, since the left is 10ft and the right is 15ft, so the middle is 15 - 10 = 5ft tall? No, maybe the middle is at the same height as the left? Wait, no, the figure has a notch. Let's look at the vertical dimensions: the right part is 15ft tall, the left part is 10ft tall, and there is a horizontal part of 15ft length, 5ft width, and (10 - 5)ft height? No, I think the correct split is:

- Prism 1: \(5\times5\times10\) (left bottom)
- Prism 2: \(15\times5\times5\) (middle, between left and right, height 5ft, since 10 - 5 = 5? No, maybe the middle is \(15\times5\times10\) and the right is \(5\times5\times5\)? Wait, no. Let's calculate the volume by adding three parts:

First part: leftmost, 5ft (width) x5ft (depth) x10ft (height): \(V_1=5\times5\times10 = 250\)

Second part: middle, 15ft (length) x5ft (depth) x5ft (height) (because the height here is 10 - 5 = 5? No, the right part is 15ft tall, so the middle part's height is 15 - 10 = 5? Wait, no, the total height of the right part is 15, left is 10, so the middle horizontal part is 15ft long, 5ft wide, and 5ft tall (15 - 10). Then the right vertical part is 5ft x5ft x15ft.

Third part: right, 5ft x5ft x15ft: \(V_3 = 5\times5\times15=375\)

Second part: middle, 15ft x5ft x5ft: \(V_2=15\times5\times5 = 375\)

Total volume: \(250+375 + 375=1000\)? But wait, the options have 875, 750, 1000, 1125. Wait, maybe my split is wrong.

Alternative approach: Let's consider the figure as a combination of two rectangular prisms.

First, the larger part: length \(5 + 15=20\) ft, width 5ft, height 10ft. Volume \(V_1=20\times5\times10 = 1000\). Then, the upper right part: length 5ft, width 5ft, height \(15 - 10 = 5\) ft. Volume \(V_2=5\times5\times5 = 125\). Total volume \(V = 1000+125 = 1125\)? No, that's not right. Wait, no, the upper right part is attached to the larger part? Wait, no, the figure is like an L - shape with an extension. Wait, let's look at the dimensions again.

Wait, the figure:

- The right - most part: 5ft (width) x5ft (depth) x15ft (height)
- The middle horizontal part: 15ft (length) x5ft (depth) x5ft (height) (since the left part is 10ft tall, and the right is 15ft, so the middle horizontal part is 15 - 10 = 5ft tall)
- The left - most part: 5ft (width) x5ft (depth) x10ft (height)

Wait, no, the depth is 5ft for all? Let's check the depth: all parts have depth 5ft.

So:

Left: \(5\times5\times10 = 250\)

Middle: \(15\times5\times5 = 375\) (because the height here is 5ft, as the right is 15 and left is 10, so the middle is 15 - 10 = 5)

Right: \(5\times5\times15 = 375\)

Total: \(250+375 + 375=1000\). But wait, the option 1000 is there. But wait, maybe I made a mistake. Wait, another way: the figure can be seen as a rectangular prism of \( (5 + 15+5)\times5\times10\) minus a rectangular prism of \(15\times5\times5\). Wait, \( (25)\times5\times10=1250\), minus \(15\times5\times5 = 375\), \(1250 - 375 = 875\). Oh! That's a better approach.

Ah, here's the mistake: The total length is \(5 + 15+5 = 25\) ft, width 5ft, height 10ft. But there is a notch of \(15\times5\times5\) (since the height of the notch is \(15 - 10 = 5\) ft? No, wait, the height of the main block is 10ft, and the right part is 15ft, so the notch is actually a missing part? Wait, no, the figure is constructed by adding the right part (5x5x15) to the main block (20x5x10). Wait, no, the main block is (5 + 15)x5x10 = 20x5x10 = 1000, and then we add the right part? No, the right part is 5x5x15, but the overlapping part is 5x5x10. So the volume is \(1000+(5\times5\times(15 - 10))=1000 + 125=1125\). No, this is confusing.

Wait, let's use the correct method for composite figures: divide into non - overlapping rectangular prisms.

Looking at the figure:

1. Bottom layer:
- Left: 5ft (length) x5ft (width) x10ft (height)
- Middle: 15ft (length) x5ft (width) x10ft (height)
- Right: 5ft (length) x5ft (width) x10ft (height)
But then there is an upper layer on the right: 5ft (length) x5ft (width) x(15 - 10)ft (height) = 5x5x5 = 125

Wait, no, the middle part's height is 10ft, left is 10ft, right is 15ft. So the bottom layer (height 10ft) has length \(5 + 15+5 = 25\) ft, width 5ft, height 10ft: volume \(25\times5\times10 = 1250\). Then, the upper layer (height 5ft) is only on the right part: length 5ft, width 5ft, height 5ft: volume \(5\times5\times5 = 125\). But that's not right. Wait, no, the middle part (15ft length) in the bottom layer: is it really 10ft tall? No, the left part is 10ft tall, the middle part (15ft length) is 5ft tall (since the right part is 15ft tall, so the middle part is 15 - 10 = 5ft tall in the vertical direction).

Ah! Now I see. The figure has three parts:

- Left: 5x5x10 (height 10, length 5, width 5)
- Middle: 15x5x5 (height 5, length 15, width 5) (because the height here is 15 - 10 = 5, since the right part is 15 and left is 10)
- Right: 5x5x15 (height 15, length 5, width 5)

Now, calculate each volume:

Left: \(5\times5\times10 = 250\)

Middle: \(15\times5\times5 = 375\)

Right: \(5\times5\times15 = 375\)

Total: \(250+375 + 375=1000\). But wait, the option 1000 is present. But let's check another way.

Wait, the figure can be considered as a rectangular prism of \(5\times5\times(10 + 15)\) minus a rectangular prism of \(15\times5\times10\). Wait, \(5\times5\times25=625\), no, that's wrong.

Wait, no, let's use the coordinates. Let's assume the bottom - left corner is (0,0,0).

- Left prism: from (0,0,0) to (5,5,10). Volume: \(5\times5\times10 = 250\)
- Middle prism: from (5,0,10) to (20,5,15). Volume: \(15\times5\times5 = 375\) (length 15, width 5, height 5)
- Right prism: from (20,0,0) to (25,5,15). Volume: \(5\times5\times15 = 375\)

Total volume: \(250 + 375+375 = 1000\). So the volume is 1000 cubic feet.

Answer:

1,000 cubic feet