Question

this is her design, using one rectangular prism and two parallelogram prisms. all prisms are one inch thick.
how many cubic inches of wood will she use in her design?
note: assume that karly will not waste any wood.
72 cubic inches
24 cubic inches
12 cubic inches

Explanation:

Step1: Calculate volume of rectangular prism

The formula for the volume of a rectangular prism is \( V = l \times w \times h \). For the rectangular prism, \( l = 6 \) in, \( w = 2 \) in, \( h = 1 \) in. So \( V_{rectangular} = 6 \times 2 \times 1 = 12 \) cubic inches.

Step2: Calculate volume of one parallelogram prism

The volume of a prism is also \( V = \text{base area} \times \text{height (thickness)} \). For a parallelogram prism, the base is a parallelogram with base \( b = 3 \) in, height \( h_{base} = 2 \) in (wait, no, looking at the diagram, the length of the parallelogram's side is 3 in, and the other dimension: wait, the thickness is 1 in, and the area of the parallelogram face: the base of the parallelogram is 3 in, height (the distance between the two parallel sides) is 2 in? Wait, no, the diagram shows for the parallelogram prisms, the dimensions: one side is 3 in, the other is 2 in? Wait, no, let's re - examine. The two parallelogram prisms: each has a base (the parallelogram) with area calculated as follows. Wait, actually, the volume of a prism is \( V=\text{area of the base} \times \text{thickness} \). For the parallelogram, the area of the base (the parallelogram face) is \( \text{base length} \times \text{height of the parallelogram} \). From the diagram, the base length of the parallelogram is 3 in, and the height (the distance perpendicular to the base length) is 2 in? Wait, no, the thickness is 1 in. Wait, actually, looking at the diagram, each parallelogram prism: the length along the non - thickness direction: one has height 3 in, and the other dimension 2 in? Wait, maybe a better way: the two parallelogram prisms are identical? Wait, no, the diagram shows two parallelogram prisms. Wait, the rectangular prism has dimensions 6x2x1. The two parallelogram prisms: each has a base (the parallelogram) with area \( 3\times2 \)? Wait, no, the thickness is 1 in. Wait, let's recast. The volume of a prism is \( V = \text{area of the polygonal base} \times \text{length (thickness)} \). For the parallelogram prisms, the base is a parallelogram with base \( b = 3 \) in, height \( h = 2 \) in (the height of the parallelogram), and the thickness (the length along the direction perpendicular to the parallelogram face) is 1 in. So the area of the parallelogram base is \( 3\times2 = 6 \)? Wait, no, that can't be. Wait, no, the height of the parallelogram: looking at the diagram, the vertical dashed lines: the total height from top to bottom of the parallelogram part is 3 + 3 = 6? No, wait, the diagram shows for the parallelogram prisms, the length of the side is 3 in, and the other dimension (the one parallel to the 2 in of the rectangular prism) is 2 in. Wait, maybe I made a mistake. Wait, the two parallelogram prisms: each has a volume. Let's look at the dimensions again. The thickness is 1 in for all prisms. For the parallelogram prisms, the base (the face that is a parallelogram) has a length of 3 in and a height (the distance between the two parallel sides) of 2 in? No, the area of the parallelogram is \( \text{base} \times \text{height} \), where the height is the perpendicular distance between the two parallel sides. From the diagram, the base of the parallelogram is 3 in, and the height (perpendicular to the base) is 2 in. So the area of the parallelogram base is \( 3\times2 = 6 \)? No, that would make the volume too big. Wait, no, the thickness is 1 in, so the volume of one parallelogram prism would be \( 3\times2\times1 = 6 \)? But that can't be. Wait, no, maybe the base of the parallelogram is 2 in and the height is 3 in? Wait, let's start over.

Wait, the formula for the volume of any prism is \( V=\text{area of the base} \times \text{height (the length along the axis perpendicular to the base)} \). For the rectangular prism, the base is a rectangle with area \( 6\times2 = 12 \), and the height (thickness) is 1, so volume 12. For the parallelogram prisms: each has a base (the parallelogram) with area \( 3\times2 = 6 \)? No, that's not right. Wait, looking at the diagram, the two parallelogram prisms: each has a length of 3 in, a width of 2 in, and a thickness of 1 in? Wait, no, the thickness is 1 in. Wait, maybe the two parallelogram prisms are each with dimensions 3x2x1? Wait, no, if we calculate the volume of one parallelogram prism: \( V_{parallelogram}=3\times2\times1 = 6 \)? But there are two of them. Wait, but that would be 12 for two. Then total volume would be 12 (rectangular) + 12 (two parallelograms) = 24? Wait, no, wait the rectangular prism volume is 6*2*1 = 12. Then each parallelogram prism: the base is a parallelogram with base 3 in, height 2 in, and thickness 1 in. So area of the parallelogram base is 3*2 = 6? No, that's not. Wait, maybe the height of the parallelogram is 2 in, and the base is 3 in, so area is 3*2 = 6, volume is 6*1 = 6. Then two parallelogram prisms: 2*6 = 12. Then total volume: 12 (rectangular) + 12 (two parallelograms) = 24? Wait, but let's check the answer options. The options are 72, 24, 12. Wait, maybe I messed up the rectangular prism. Wait, the rectangular prism: length 6, width 2, height 1. Volume: 6*2*1 = 12. The two parallelogram prisms: each has a base (the parallelogram) with area 3*2 = 6? No, wait, the height of the parallelogram: looking at the diagram, the vertical dashed line is 3 in, and the other dimension is 2 in. Wait, maybe the area of the parallelogram is 3*2 = 6, and thickness 1, so volume 6 per parallelogram. Two of them: 12. Then total volume: 12 + 12 = 24. Wait, but let's re - calculate.

Wait, the rectangular prism: \( V = 6\times2\times1=12 \).

Each parallelogram prism: the base is a parallelogram. The formula for the area of a parallelogram is \( A = b\times h \), where \( b \) is the base and \( h \) is the height (perpendicular to the base). From the diagram, the base \( b = 3 \) in, and the height \( h = 2 \) in. So area \( A = 3\times2 = 6 \) square inches. The volume of a prism is \( V = A\times \text{thickness} \). The thickness is 1 in, so volume of one parallelogram prism is \( 6\times1 = 6 \) cubic inches. There are two parallelogram prisms, so their total volume is \( 2\times6 = 12 \) cubic inches.

Now, total volume of the design is the sum of the volume of the rectangular prism and the two parallelogram prisms: \( V_{total}=12 + 12=24 \) cubic inches.

Answer:

24 cubic inches