Question

riving a formula for volume of a pyramid
the area of the base of the cube, ( b ), is ((b)(b)) square units.
the volume of the cube is ((b)(b)(b)) cubic units.
the height of each pyramid, ( h ), is (\boldsymbol{\text{dropdown}}). therefore, ( b = 2h ).
there are (\boldsymbol{\text{dropdown}}) square pyramids with the same base and height that exactly fill the given cube.
therefore, the volume of one pyramid is (\boldsymbol{\text{dropdown}}) or (\frac{1}{3}bh).

Explanation:

Step1: Analyze the height of the pyramid

From the equation \( b = 2h \), we can solve for \( h \) by dividing both sides by 2. So \( h=\frac{b}{2} \). So the height of each pyramid is \( \frac{b}{2} \).

Step2: Determine the number of pyramids in the cube

Looking at the cube and the pyramids formed, we can see that there are 6 square pyramids? Wait, no, actually, in a cube, if we divide it into pyramids with the same base and height, the number of such pyramids that fill the cube: Let's think about the volume of the cube \( V = b^3 \) and the volume of one pyramid. But from the diagram, we can see that the cube is divided into 6 pyramids? Wait, no, actually, the standard way: a cube can be divided into 6 square pyramids? Wait, no, let's check the volume. The volume of the cube is \( b^3 \). If we have pyramids with base area \( B = b^2 \) and height \( h=\frac{b}{2} \), but wait, the formula for the volume of a pyramid is \( \frac{1}{3}Bh \). Wait, in the cube, if we consider the pyramids, actually, the cube can be divided into 6 pyramids? No, wait, let's look at the diagram. The cube has side length \( b \), and the height of each pyramid is \( \frac{b}{2} \) (since \( b = 2h \)). Now, how many pyramids? Let's see, the cube's volume is \( b^3 \). The volume of one pyramid is \( \frac{1}{3}Bh=\frac{1}{3}b^2\times\frac{b}{2}=\frac{b^3}{6} \). Then the number of pyramids would be \( \frac{b^3}{\frac{b^3}{6}} = 6 \)? Wait, no, maybe I made a mistake. Wait, the problem says "There are [ ] square pyramids with the same base and height that exactly fill the given cube." Wait, looking at the diagram, the cube is divided into 6 pyramids? Wait, no, actually, the correct number is 6? Wait, no, let's check the standard. Wait, a cube can be divided into 6 square pyramids, each with base area \( b^2 \) and height \( \frac{b}{2} \). But maybe in this case, the number is 6? Wait, no, let's re - examine. The volume of the cube is \( b^3 \). The volume of one pyramid is \( \frac{1}{3}Bh \), and since \( B = b^2 \) and \( h=\frac{b}{2} \), then \( V_{pyramid}=\frac{1}{3}\times b^2\times\frac{b}{2}=\frac{b^3}{6} \). Then the number of pyramids \( n=\frac{V_{cube}}{V_{pyramid}}=\frac{b^3}{\frac{b^3}{6}} = 6 \). But wait, maybe the diagram shows 6? Wait, no, the options? Wait, maybe I misread. Wait, the problem's next part: "Therefore, the volume of one pyramid is [ ] or \( \frac{1}{3}Bh \)". Let's see, the volume of the cube is \( b^3 \), and if there are 6 pyramids, then the volume of one pyramid is \( \frac{b^3}{6} \), but also, since \( B = b^2 \) and \( h=\frac{b}{2} \), \( \frac{1}{3}Bh=\frac{1}{3}\times b^2\times\frac{b}{2}=\frac{b^3}{6} \), which matches. But maybe the number of pyramids is 6? Wait, no, maybe the diagram is different. Wait, looking at the cube, the center point, and the pyramids: actually, the cube can be divided into 6 square pyramids, each with a face of the cube as the base and the center of the cube as the apex. So the number of pyramids is 6? Wait, but let's check the height. The distance from the center of the cube to a face is \( \frac{b}{2} \), which is the height of each pyramid. So the number of pyramids is 6. But wait, the problem's dropdown for the number of pyramids: maybe I made a mistake. Wait, let's think again. The volume of the cube is \( b^3 \). The volume of one pyramid is \( \frac{1}{3}Bh \), and if \( B = b^2 \) and \( h=\frac{b}{2} \), then \( \frac{1}{3}b^2\times\frac{b}{2}=\frac{b^3}{6} \). So the number of pyramids is \( \frac{b^3}{\frac{b^3}{6}} = 6 \). But maybe the problem has a different approach. Wait, the problem says "There are [ ] square pyramids with the same base and height that exactly fill the given cube." Let's check the options (even though not shown, but from the context). Wait, maybe the number is 6? No, wait, maybe I messed up. Wait, the standard formula derivation: a cube can be divided into 6 square pyramids? No, actually, a square prism (cube is a square prism) can be divided into 3 square pyramids with the same base and height. Wait, that's the standard: the volume of a pyramid is one - third the volume of a prism with the same base and height. Wait, so if we have a cube (a square prism with side \( b \)), then the volume of the prism (cube) is \( b^3 \). If we divide it into 3 pyramids with the same base ( \( b\times b \)) and height ( \( b \) ), then each pyramid has volume \( \frac{1}{3}b^3 \). But in this diagram, the height of the pyramid is \( \frac{b}{2} \), not \( b \). Wait, the diagram shows that the height \( h \) of the pyramid is such that \( b = 2h \), so \( h=\frac{b}{2} \). So the base of the pyramid is a square with side \( b \), and the height is \( \frac{b}{2} \). Then the volume of one pyramid is \( \frac{1}{3}\times b^2\times\frac{b}{2}=\frac{b^3}{6} \). The volume of the cube is \( b^3 \), so the number of pyramids is \( \frac{b^3}{\frac{b^3}{6}} = 6 \). But maybe the problem is considering a different division. Wait, the problem's next part: "Therefore, the volume of one pyramid is [ ] or \( \frac{1}{3}Bh \)". Let's see, the volume of the cube is \( b^3 \), and if there are 6 pyramids, then the volume of one pyramid is \( \frac{b^3}{6} \), and since \( B = b^2 \) and \( h=\frac{b}{2} \), \( \frac{1}{3}Bh=\frac{1}{3}b^2\times\frac{b}{2}=\frac{b^3}{6} \), which matches. But maybe the number of pyramids is 6? Wait, no, maybe the diagram is dividing the cube into 6 pyramids? Wait, maybe I was wrong earlier. Let's check the height again. The height of the pyramid is \( \frac{b}{2} \) (from \( b = 2h \)). The base area is \( b\times b \). So volume of pyramid is \( \frac{1}{3}\times base\ area\times height=\frac{1}{3}\times b^2\times\frac{b}{2}=\frac{b^3}{6} \). Volume of cube is \( b^3 \), so number of pyramids is \( \frac{b^3}{\frac{b^3}{6}} = 6 \). But maybe the problem has a different answer. Wait, maybe the number of pyramids is 6? But let's go back to the problem.

Wait, the problem's first dropdown for height: we found \( h = \frac{b}{2} \). Then the number of pyramids: let's see, the cube is divided into 6 pyramids? Or maybe 3? Wait, no, the standard formula for the volume of a pyramid is \( \frac{1}{3}Bh \), where \( B \) is the base area and \( h \) is the height of the pyramid. In the case of a cube, if we have a pyramid with base equal to the face of the cube (area \( B = b^2 \)) and height equal to the edge length of the cube ( \( h = b \) ), then the volume of the pyramid is \( \frac{1}{3}b^3 \), and the number of such pyramids in the cube is 3 (since \( 3\times\frac{1}{3}b^3=b^3 \)). But in this diagram, the height of the pyramid is \( \frac{b}{2} \), not \( b \). So that's a different case. So in this diagram, since \( b = 2h \), \( h=\frac{b}{2} \). Then the volume of one pyramid is \( \frac{1}{3}\times b^2\times\frac{b}{2}=\frac{b^3}{6} \), and the number of pyramids is \( \frac{b^3}{\frac{b^3}{6}} = 6 \). But maybe the problem is considering a different division. Wait, the problem says "There are [ ] square pyramids with the same base and height that exactly fill the given cube." Let's assume that the answer for the number of pyramids is 6? No, wait, maybe I made a mistake. Let's look at the diagram again. The cube has side length \( b \), and the pyramids are formed with the center of the cube as the apex. So each pyramid has a triangular face? No, square base. Wait, the base of each pyramid is a square face of the cube, and the height is the distance from the center of the cube to that face, which is \( \frac{b}{2} \). So there are 6 faces, so 6 pyramids. But the volume of each pyramid is \( \frac{1}{3}\times b^2\times\frac{b}{2}=\frac{b^3}{6} \), and 6 of them give \( 6\times\frac{b^3}{6}=b^3 \), which is the volume of the cube. So that works.

Then, the volume of one pyramid: since the volume of the cube is \( b^3 \) and there are 6 pyramids, the volume of one pyramid is \( \frac{b^3}{6} \). But also, since \( B = b^2 \) and \( h=\frac{b}{2} \), \( \frac{1}{3}Bh=\frac{1}{3}\times b^2\times\frac{b}{2}=\frac{b^3}{6} \), which matches.

Step3: Fill in the blanks

- Height of each pyramid: \( \frac{b}{2} \) (since \( b = 2h\Rightarrow h=\frac{b}{2} \))
- Number of pyramids: 6? Wait, no, maybe the problem has a different number. Wait, maybe I was wrong. Wait, the standard derivation of the volume of a pyramid: a square prism (cube) can be divided into 3 square pyramids with the same base and height. But in that case, the height of the pyramid is equal to the height of the prism (cube). But in this diagram, the height of the pyramid is half the side length of the cube. So maybe this is a different derivation. Wait, let's re - examine the problem.

Wait, the problem says "the height of each pyramid, \( h \), is [ ]" and we have \( b = 2h \), so \( h=\frac{b}{2} \), so the height is \( \frac{b}{2} \).

Then, "There are [ ] square pyramids with the same base and height that exactly fill the given cube." Let's calculate the volume of one pyramid: \( V_{pyramid}=\frac{1}{3}Bh \), and \( B = b^2 \), \( h=\frac{b}{2} \), so \( V_{pyramid}=\frac{1}{3}\times b^2\times\frac{b}{2}=\frac{b^3}{6} \). The volume of the cube is \( V_{cube}=b^3 \). So the number of pyramids \( n=\frac{V_{cube}}{V_{pyramid}}=\frac{b^3}{\frac{b^3}{6}} = 6 \). So there are 6 pyramids.

Then, "Therefore, the volume of one pyramid is [ ] or \( \frac{1}{3}Bh \)". The volume of one pyramid is \( \frac{b^3}{6} \), and since \( B = b^2 \) and \( h=\frac{b}{2} \), \( \frac{1}{3}Bh=\frac{1}{3}\times b^2\times\frac{b}{2}=\frac{b^3}{6} \), so the volume of one pyramid is \( \frac{b^3}{6} \) (or \( \frac{1}{3}Bh \)).

Answer:

• Height of each pyramid: \( \frac{b}{2} \)
• Number of pyramids: 6
• Volume of one pyramid: \( \frac{b^3}{6} \)