QUESTION IMAGE
Question
fluency and skills practice
breaking apart figures
to find volume continued
(diagram with dimensions: 2 cm, 2 cm, 2 cm, 5 cm, 3 cm, 9 cm, and handwritten notes)
To find the volume of the composite figure, we can break it into simpler rectangular prisms. Let's assume the figure has a base length of \( 9 \, \text{cm} \), height of \( 3 \, \text{cm} \), and depth of \( 5 \, \text{cm} \) (from the \( 5 \, \text{cm} \) markings). Additionally, there are two smaller prisms on the sides. Wait, maybe a better approach is to consider the main rectangular prism and the two side prisms. But looking at the dimensions, maybe the figure is a combination where we can calculate the volume as follows:
Step 1: Identify the dimensions of the main part
The main rectangular prism (the lower part) has length \( 9 \, \text{cm} \), width \( 5 \, \text{cm} \), and height \( 3 \, \text{cm} \). The volume of a rectangular prism is given by \( V = l \times w \times h \).
Step 2: Calculate the volume of the main part
For the main part:
\( V_{\text{main}} = 9 \times 5 \times 3 \)
\( V_{\text{main}} = 135 \, \text{cm}^3 \)
Wait, but there are also the two side prisms. Wait, maybe the figure is a bit more complex. Wait, the \( 2 \, \text{cm} \) markings: maybe the total height is \( 3 + 2 = 5 \, \text{cm} \)? Wait, no, the \( 3 \, \text{cm} \) is marked. Wait, maybe the figure is a combination where we have a base of \( 9 \, \text{cm} \), width \( 5 \, \text{cm} \), and total height \( 5 \, \text{cm} \), but with a middle section removed? Wait, no, the problem says "Breaking Apart Figures to Find Volume". Let's re-examine the diagram.
Looking at the diagram, there are three parts? Wait, maybe the figure is composed of three rectangular prisms: two on the sides and one in the middle. Wait, the length of the middle part: the total length is \( 9 \, \text{cm} \), and the two side parts each have a length of \( 2 \, \text{cm} \), so the middle part's length is \( 9 - 2 - 2 = 5 \, \text{cm} \). Wait, no, maybe the height: the lower part is \( 3 \, \text{cm} \) tall, and the upper parts (the blue ones) are \( 2 \, \text{cm} \) tall? Wait, the \( 5 \, \text{cm} \) is the depth (width) of the figure.
Wait, maybe the correct approach is to consider the entire figure as a rectangular prism with length \( 9 \, \text{cm} \), width \( 5 \, \text{cm} \), and height \( 5 \, \text{cm} \) (since \( 3 + 2 = 5 \)), but that might not be right. Wait, the diagram has \( 5 \, \text{cm} \) (depth), \( 9 \, \text{cm} \) (length), \( 3 \, \text{cm} \) (height of the lower part), and \( 2 \, \text{cm} \) (height of the upper parts). Wait, maybe the figure is made up of three rectangular prisms:
- Left prism: length \( 2 \, \text{cm} \), width \( 5 \, \text{cm} \), height \( 5 \, \text{cm} \) (since \( 3 + 2 = 5 \))
- Middle prism: length \( 5 \, \text{cm} \) (since \( 9 - 2 - 2 = 5 \)), width \( 5 \, \text{cm} \), height \( 3 \, \text{cm} \)
- Right prism: length \( 2 \, \text{cm} \), width \( 5 \, \text{cm} \), height \( 5 \, \text{cm} \)
Let's calculate each volume:
Step 1: Volume of left prism
\( V_{\text{left}} = 2 \times 5 \times 5 = 50 \, \text{cm}^3 \)
Step 2: Volume of middle prism
\( V_{\text{middle}} = 5 \times 5 \times 3 = 75 \, \text{cm}^3 \)
Step 3: Volume of right prism
\( V_{\text{right}} = 2 \times 5 \times 5 = 50 \, \text{cm}^3 \)
Step 4: Total volume
\( V_{\text{total}} = V_{\text{left}} + V_{\text{middle}} + V_{\text{right}} \)
\( V_{\text{total}} = 50 + 75 + 50 = 175 \, \text{cm}^3 \)
Wait, but the handwritten note has \( 175 \), so maybe that's the answer. Alternatively, if we consider the main part as \( 9 \times 5 \times 3 = 135 \) and the two side parts as \( 2 \times 5 \times 2 \) each…
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The volume of the figure is \(\boxed{175}\) cubic centimeters.