Question

fluency and skills practice
lesson 3
breaking apart figures to find
7
2 cm
2 cm
2 cm
5 cm
5 cm
9 cm

Explanation:

Step1: Calculate volume of large rectangular prism

The large rectangular prism has length \( l = 9 \, \text{cm} \), width \( w = 5 \, \text{cm} \), height \( h = 5 \, \text{cm} \). Volume formula: \( V_{\text{large}} = l \times w \times h \)
\( V_{\text{large}} = 9 \times 5 \times 5 = 225 \, \text{cm}^3 \)

Step2: Calculate volume of the removed smaller rectangular prism

The removed part has length \( l = 2 \, \text{cm} \) (since \( 9 - 2 - 2 = 5 \)? Wait, no, the top indent: length is \( 9 - 2 - 2 = 5 \)? Wait, the figure shows the indent has length (the horizontal part) as \( 9 - 2 - 2 = 5 \)? Wait, no, the indent's dimensions: the height of the indent is \( 2 \, \text{cm} \), width is \( 5 \, \text{cm} \), length is \( 2 \, \text{cm} \)? Wait, no, looking at the diagram: the base is \( 9 \, \text{cm} \) (length), \( 5 \, \text{cm} \) (width), height \( 5 \, \text{cm} \). The indent is a rectangular prism with length \( (9 - 2 - 2) = 5 \)? Wait, no, the top has two \( 2 \, \text{cm} \) parts, so the middle indent's length is \( 9 - 2 - 2 = 5 \)? Wait, no, the indent's length (along the base) is \( 9 - 2 - 2 = 5 \)? Wait, no, the diagram shows the indent has a height of \( 2 \, \text{cm} \), width of \( 5 \, \text{cm} \), and length of \( 2 \, \text{cm} \)? Wait, maybe I misread. Wait, the figure: the main block is \( 9 \, \text{cm} \) (length), \( 5 \, \text{cm} \) (width), \( 5 \, \text{cm} \) (height). The indent is a rectangular prism with length \( (9 - 2 - 2) = 5 \)? No, the horizontal dimension: the top has two \( 2 \, \text{cm} \) sections, so the middle indent's length (the horizontal part) is \( 9 - 2 - 2 = 5 \)? Wait, no, the indent's length (the side parallel to the \( 9 \, \text{cm} \) side) is \( 5 \, \text{cm} \)? Wait, no, the width is \( 5 \, \text{cm} \) (same as the base width), height of indent is \( 2 \, \text{cm} \), and length (the front-back) is \( 2 \, \text{cm} \)? Wait, no, the diagram: the indent is a rectangular prism with length \( 5 \, \text{cm} \) (since \( 9 - 2 - 2 = 5 \)), width \( 5 \, \text{cm} \), height \( 2 \, \text{cm} \)? Wait, no, the height of the indent is \( 2 \, \text{cm} \), width is \( 5 \, \text{cm} \), length is \( 2 \, \text{cm} \)? I think I made a mistake. Wait, the correct way: the main block is \( 9 \times 5 \times 5 \). The removed part is a rectangular prism with length \( (9 - 2 - 2) = 5 \)? No, the top indent: the length (along the 9 cm side) is \( 9 - 2 - 2 = 5 \), width is \( 5 \) (same as base width), height is \( 2 \). Wait, no, the width of the removed part is \( 5 \, \text{cm} \) (the depth), length is \( (9 - 2 - 2) = 5 \, \text{cm} \), height is \( 2 \, \text{cm} \). So volume of removed: \( V_{\text{removed}} = 5 \times 5 \times 2 = 50 \, \text{cm}^3 \)? Wait, no, that can't be. Wait, looking at the diagram again: the indent has a height of \( 2 \, \text{cm} \), the width (depth) is \( 5 \, \text{cm} \), and the length (horizontal) is \( 2 \, \text{cm} \)? Wait, the original figure: the base is \( 9 \, \text{cm} \) (length), \( 5 \, \text{cm} \) (width), height \( 5 \, \text{cm} \). The indent is a rectangular prism with length \( 2 \, \text{cm} \), width \( 5 \, \text{cm} \), height \( 2 \, \text{cm} \)? No, that doesn't fit. Wait, maybe the indent's length is \( 9 - 2 - 2 = 5 \, \text{cm} \), width \( 5 \, \text{cm} \), height \( 2 \, \text{cm} \). So \( V_{\text{removed}} = 5 \times 5 \times 2 = 50 \). Then total volume is \( 225 - 50 = 175 \)? Wait, no, that's not right. Wait, maybe the width of the removed part is \( 5 \, \text{cm} \), length is \( 2 \, \text{cm} \), height is \( 2 \, \text{cm} \). Wait, the diagram shows the indent has a horizontal length of \( 2 \, \text{cm} \)? No, the labels: the top has two \( 2 \, \text{cm} \) segments, and the indent's height is \( 2 \, \text{cm} \). So the main block: length \( 9 \), width \( 5 \), height \( 5 \). The removed part: length \( (9 - 2 - 2) = 5 \), width \( 5 \), height \( 2 \). So \( V_{\text{large}} = 9 \times 5 \times 5 = 225 \), \( V_{\text{removed}} = 5 \times 5 \times 2 = 50 \), so total volume \( 225 - 50 = 175 \, \text{cm}^3 \). Wait, but maybe I messed up the removed part's dimensions. Alternatively, break the figure into three parts: two side blocks and one middle block. The side blocks: each has length \( 2 \, \text{cm} \), width \( 5 \, \text{cm} \), height \( 5 \, \text{cm} \). The middle block: length \( (9 - 2 - 2) = 5 \, \text{cm} \), width \( 5 \, \text{cm} \), height \( (5 - 2) = 3 \, \text{cm} \). Then calculate each:

Side blocks: 2 blocks, each \( 2 \times 5 \times 5 = 50 \), so two: \( 2 \times 50 = 100 \)

Middle block: \( 5 \times 5 \times 3 = 75 \)

Total volume: \( 100 + 75 = 175 \, \text{cm}^3 \). Yes, that makes sense. So:

Step1: Volume of one side block

\( V_{\text{side}} = 2 \times 5 \times 5 = 50 \, \text{cm}^3 \)

Step2: Volume of two side blocks

\( V_{\text{two sides}} = 2 \times 50 = 100 \, \text{cm}^3 \)

Step3: Volume of the middle block

Height of middle block: \( 5 - 2 = 3 \, \text{cm} \), length \( 9 - 2 - 2 = 5 \, \text{cm} \), width \( 5 \, \text{cm} \)
\( V_{\text{middle}} = 5 \times 5 \times 3 = 75 \, \text{cm}^3 \)

Step4: Total volume

\( V_{\text{total}} = V_{\text{two sides}} + V_{\text{middle}} = 100 + 75 = 175 \, \text{cm}^3 \)

Answer:

The volume of the figure is \( \boxed{175} \) cubic centimeters.