Question

breaking apart figures to find volume continued
7
2 cm
2 cm
5 cm
2 cm
5 cm
3 cm
9 cm
8
6 in.
3 in.
3 in.
8 in.
3 in.
2 in.
11 in.
9 what are two different ways to break apart the figure in problem 3 to find its volume?

Explanation:

Since the problem in question 9 refers to "the figure in problem 3" (which isn't shown here), we'll assume a general composite 3D figure (like a rectangular prism with parts added or removed) and provide two common ways to break it apart:

Way 1: Subtraction Method (Remove a Smaller Prism)

1. Identify the Outer Prism: Find the dimensions of the largest rectangular prism that encloses the composite figure. Calculate its volume using \( V_{\text{outer}} = l \times w \times h \).
2. Identify the Missing Prism: Determine the dimensions of the "missing" or removed part (e.g., a smaller rectangular prism cut out from the outer prism). Calculate its volume using \( V_{\text{missing}} = l_{\text{missing}} \times w_{\text{missing}} \times h_{\text{missing}} \).
3. Subtract Volumes: The volume of the composite figure is \( V = V_{\text{outer}} - V_{\text{missing}} \).

Way 2: Addition Method (Combine Smaller Prisms)

1. Divide into Smaller Prisms: Split the composite figure into two or more non - overlapping rectangular prisms. For example, if the figure has a stepped shape, we can split it into prisms along the steps.
2. Calculate Volume of Each Prism: For each smaller prism, use the formula \( V_i = l_i \times w_i \times h_i \), where \( i \) represents each individual prism.
3. Add Volumes: Sum up the volumes of all the smaller prisms: \( V=\sum_{i = 1}^{n}V_i\) (where \( n \) is the number of smaller prisms).

If we assume a specific example (since problem 3's figure is not given, let's take a common stepped - like composite prism with length \( L \), width \( W \), and we can split it as follows):

Example - Based Explanation (Hypothetical Figure)

Suppose the figure in problem 3 is a composite prism that can be thought of as:

• Way 1 (Subtraction): The outer prism has dimensions \( l = 10\space\text{cm}, w = 5\space\text{cm}, h = 6\space\text{cm} \), so \( V_{\text{outer}}=10\times5\times6 = 300\space\text{cm}^3 \). The missing prism has dimensions \( l = 3\space\text{cm}, w = 5\space\text{cm}, h = 2\space\text{cm} \), so \( V_{\text{missing}}=3\times5\times2=30\space\text{cm}^3 \). Then \( V = 300 - 30=270\space\text{cm}^3 \).
• Way 2 (Addition): Split it into two prisms. Prism 1: \( l = 7\space\text{cm}, w = 5\space\text{cm}, h = 6\space\text{cm} \), \( V_1 = 7\times5\times6 = 210\space\text{cm}^3 \). Prism 2: \( l = 3\space\text{cm}, w = 5\space\text{cm}, h = 4\space\text{cm} \), \( V_2=3\times5\times4 = 60\space\text{cm}^3 \). Then \( V=210 + 60 = 270\space\text{cm}^3 \).

In general, for most composite rectangular - prism - based figures (which is a common case in volume - by - breaking - apart problems):

• One way is to subtract the volume of a removed rectangular prism from the volume of a larger enclosing rectangular prism.
• The other way is to divide the composite figure into two or more non - overlapping rectangular prisms and then add their volumes together.

If we had the exact figure from problem 3, we could be more specific, but these are the two fundamental methods for finding the volume of composite figures by breaking them apart.

Answer:

Since the problem in question 9 refers to "the figure in problem 3" (which isn't shown here), we'll assume a general composite 3D figure (like a rectangular prism with parts added or removed) and provide two common ways to break it apart:

Way 1: Subtraction Method (Remove a Smaller Prism)

1. Identify the Outer Prism: Find the dimensions of the largest rectangular prism that encloses the composite figure. Calculate its volume using \( V_{\text{outer}} = l \times w \times h \).
2. Identify the Missing Prism: Determine the dimensions of the "missing" or removed part (e.g., a smaller rectangular prism cut out from the outer prism). Calculate its volume using \( V_{\text{missing}} = l_{\text{missing}} \times w_{\text{missing}} \times h_{\text{missing}} \).
3. Subtract Volumes: The volume of the composite figure is \( V = V_{\text{outer}} - V_{\text{missing}} \).

Way 2: Addition Method (Combine Smaller Prisms)

1. Divide into Smaller Prisms: Split the composite figure into two or more non - overlapping rectangular prisms. For example, if the figure has a stepped shape, we can split it into prisms along the steps.
2. Calculate Volume of Each Prism: For each smaller prism, use the formula \( V_i = l_i \times w_i \times h_i \), where \( i \) represents each individual prism.
3. Add Volumes: Sum up the volumes of all the smaller prisms: \( V=\sum_{i = 1}^{n}V_i\) (where \( n \) is the number of smaller prisms).

If we assume a specific example (since problem 3's figure is not given, let's take a common stepped - like composite prism with length \( L \), width \( W \), and we can split it as follows):

Example - Based Explanation (Hypothetical Figure)

Suppose the figure in problem 3 is a composite prism that can be thought of as:

• Way 1 (Subtraction): The outer prism has dimensions \( l = 10\space\text{cm}, w = 5\space\text{cm}, h = 6\space\text{cm} \), so \( V_{\text{outer}}=10\times5\times6 = 300\space\text{cm}^3 \). The missing prism has dimensions \( l = 3\space\text{cm}, w = 5\space\text{cm}, h = 2\space\text{cm} \), so \( V_{\text{missing}}=3\times5\times2=30\space\text{cm}^3 \). Then \( V = 300 - 30=270\space\text{cm}^3 \).
• Way 2 (Addition): Split it into two prisms. Prism 1: \( l = 7\space\text{cm}, w = 5\space\text{cm}, h = 6\space\text{cm} \), \( V_1 = 7\times5\times6 = 210\space\text{cm}^3 \). Prism 2: \( l = 3\space\text{cm}, w = 5\space\text{cm}, h = 4\space\text{cm} \), \( V_2=3\times5\times4 = 60\space\text{cm}^3 \). Then \( V=210 + 60 = 270\space\text{cm}^3 \).

In general, for most composite rectangular - prism - based figures (which is a common case in volume - by - breaking - apart problems):

• One way is to subtract the volume of a removed rectangular prism from the volume of a larger enclosing rectangular prism.
• The other way is to divide the composite figure into two or more non - overlapping rectangular prisms and then add their volumes together.

If we had the exact figure from problem 3, we could be more specific, but these are the two fundamental methods for finding the volume of composite figures by breaking them apart.