Question

tori is creating an informal argument for the formula for the volume of a square pyramid. to start, she considered a cube with side lengths of 10 units. then, tori sliced through the center of the cube along its diagonals, creating identical square pyramids. toris diagram is shown below. how many square pyramids did tori slice the cube into? what is the height of each square pyramid? units pick all the expressions that can be used to represent the volume of the square pyramid. ( \frac{1}{5}(10 cdot 10 cdot 10) ) ( \frac{1}{6}(10 cdot 10 cdot 10) ) ( \frac{4}{3}(10 cdot 10 cdot 10) ) ( \frac{1}{3}(10 cdot 10)(5) ) ( \frac{8}{3}(10 cdot 10)(5) ) ( \frac{1}{2}(10 cdot 10)(6) )

Explanation:

Step1: Determine the number of square pyramids

A cube has 6 faces. When slicing through the center of the cube along its diagonals, we get 6 identical square pyramids.

Step2: Find the height of each square pyramid

The side - length of the cube is \(s = 10\) units. The height \(h\) of each square pyramid is half of the side - length of the cube. So \(h=\frac{10}{2}=5\) units.

Step3: Calculate the volume of the cube

The volume of a cube \(V_{cube}=s\times s\times s\), where \(s = 10\) units. So \(V_{cube}=10\times10\times10\)

Step4: Calculate the volume of one square pyramid

The volume of a square pyramid \(V=\frac{1}{3}Bh\), where \(B = s\times s=10\times10\) (base area) and \(h = 5\) (height). So \(V=\frac{1}{3}(10\times10)\times5\). Also, since the volume of the cube is \(V_{cube}=10\times10\times10\) and there are 6 pyramids, the volume of one pyramid \(V=\frac{1}{6}(10\times10\times10)\)

Answer:

• The number of square pyramids: \(6\)
• The height of each square pyramid: \(5\) units
• The expressions for the volume of the square pyramid: \(\frac{1}{6}(10\cdot10\cdot10)\) and \(\frac{1}{3}(10\cdot10)(5)\)