Question

find the area and volume of both prisms.

Explanation:

To solve for the surface area and volume of the rectangular prisms, we first identify the dimensions (length \( l \), width \( w \), height \( h \)) by counting the unit cubes.

Left Prism:

Count the unit cubes:

• Length (\( l \)): 6 units (number of cubes along the length).
• Width (\( w \)): 4 units (number of cubes along the width).
• Height (\( h \)): 2 units (number of cubes along the height).

Step 1: Surface Area of a Rectangular Prism

The formula for the surface area (\( SA \)) of a rectangular prism is:
\[ SA = 2(lw + lh + wh) \]

Substitute \( l = 6 \), \( w = 4 \), \( h = 2 \):
\[

$$\begin{align*} SA &= 2[(6 \times 4) + (6 \times 2) + (4 \times 2)] \\ &= 2[24 + 12 + 8] \\ &= 2[44] \\ &= 88 \text{ square units} \end{align*}$$

\]

Step 2: Volume of a Rectangular Prism

The formula for the volume (\( V \)) of a rectangular prism is:
\[ V = l \times w \times h \]

Substitute \( l = 6 \), \( w = 4 \), \( h = 2 \):
\[

$$\begin{align*} V &= 6 \times 4 \times 2 \\ &= 48 \text{ cubic units} \end{align*}$$

\]

Right Prism:

Count the unit cubes:

• Length (\( l \)): 7 units (number of cubes along the length).
• Width (\( w \)): 3 units (number of cubes along the width).
• Height (\( h \)): 7 units (number of cubes along the height).

Step 1: Surface Area of a Rectangular Prism

Using the surface area formula \( SA = 2(lw + lh + wh) \):

Substitute \( l = 7 \), \( w = 3 \), \( h = 7 \):
\[

$$\begin{align*} SA &= 2[(7 \times 3) + (7 \times 7) + (3 \times 7)] \\ &= 2[21 + 49 + 21] \\ &= 2[91] \\ &= 182 \text{ square units} \end{align*}$$

\]

Step 2: Volume of a Rectangular Prism

Using the volume formula \( V = l \times w \times h \):

Substitute \( l = 7 \), \( w = 3 \), \( h = 7 \):
\[

$$\begin{align*} V &= 7 \times 3 \times 7 \\ &= 147 \text{ cubic units} \end{align*}$$

\]

Final Answers:

• Left Prism:

Surface Area: \( \boldsymbol{88} \) square units, Volume: \( \boldsymbol{48} \) cubic units.

• Right Prism:

Surface Area: \( \boldsymbol{182} \) square units, Volume: \( \boldsymbol{147} \) cubic units.

Answer:

To solve for the surface area and volume of the rectangular prisms, we first identify the dimensions (length \( l \), width \( w \), height \( h \)) by counting the unit cubes.

Left Prism:

Count the unit cubes:

• Length (\( l \)): 6 units (number of cubes along the length).
• Width (\( w \)): 4 units (number of cubes along the width).
• Height (\( h \)): 2 units (number of cubes along the height).

Step 1: Surface Area of a Rectangular Prism

The formula for the surface area (\( SA \)) of a rectangular prism is:
\[ SA = 2(lw + lh + wh) \]

Substitute \( l = 6 \), \( w = 4 \), \( h = 2 \):
\[

$$\begin{align*} SA &= 2[(6 \times 4) + (6 \times 2) + (4 \times 2)] \\ &= 2[24 + 12 + 8] \\ &= 2[44] \\ &= 88 \text{ square units} \end{align*}$$

\]

Step 2: Volume of a Rectangular Prism

The formula for the volume (\( V \)) of a rectangular prism is:
\[ V = l \times w \times h \]

Substitute \( l = 6 \), \( w = 4 \), \( h = 2 \):
\[

$$\begin{align*} V &= 6 \times 4 \times 2 \\ &= 48 \text{ cubic units} \end{align*}$$

\]

Right Prism:

Count the unit cubes:

• Length (\( l \)): 7 units (number of cubes along the length).
• Width (\( w \)): 3 units (number of cubes along the width).
• Height (\( h \)): 7 units (number of cubes along the height).

Step 1: Surface Area of a Rectangular Prism

Using the surface area formula \( SA = 2(lw + lh + wh) \):

Substitute \( l = 7 \), \( w = 3 \), \( h = 7 \):
\[

$$\begin{align*} SA &= 2[(7 \times 3) + (7 \times 7) + (3 \times 7)] \\ &= 2[21 + 49 + 21] \\ &= 2[91] \\ &= 182 \text{ square units} \end{align*}$$

\]

Step 2: Volume of a Rectangular Prism

Using the volume formula \( V = l \times w \times h \):

Substitute \( l = 7 \), \( w = 3 \), \( h = 7 \):
\[

$$\begin{align*} V &= 7 \times 3 \times 7 \\ &= 147 \text{ cubic units} \end{align*}$$

\]

Final Answers:

• Left Prism:

Surface Area: \( \boldsymbol{88} \) square units, Volume: \( \boldsymbol{48} \) cubic units.

• Right Prism:

Surface Area: \( \boldsymbol{182} \) square units, Volume: \( \boldsymbol{147} \) cubic units.