To solve this problem, we need to determine the volumes of both Michelle's cube and Tasha's pyramid.
Step 1: Calculate the volume of Michelle's cube.
The formula for the volume \( V \) of a cube is given by: \[ V = s^3 \] where \( s \) is the length of one side of the cube.
Since Michelle's cube has a height of 6 cm, we know that all sides of the cube are equal, so: \[ s = 6 , \text{cm} \]
Now we can find the volume: \[ V = 6^3 = 6 \times 6 \times 6 = 216 , \text{cm}^3 \]
Step 2: Calculate the volume of Tasha's pyramid.
The formula for the volume \( V \) of a square-based pyramid is given by: \[ V = \frac{1}{3} \times B \times h \] where \( B \) is the area of the base and \( h \) is the height of the pyramid.
Since Tasha's pyramid has the same height and base area as Michelle's cube, we first need to determine the base area of the cube. The base area of the cube is: \[ B = s^2 \] Substituting the side length of the cube: \[ B = 6^2 = 36 , \text{cm}^2 \]
Now, we know that the height of Tasha's pyramid is also 6 cm. Now we can calculate the volume of the pyramid: \[ V = \frac{1}{3} \times B \times h = \frac{1}{3} \times 36 , \text{cm}^2 \times 6 , \text{cm} \]
Calculating the volume: \[ V = \frac{1}{3} \times 36 \times 6 = \frac{216}{3} = 72 , \text{cm}^3 \]
Final Answer: The volume of Michelle's cube is \( 216 , \text{cm}^3 \) and the volume of Tasha's pyramid is \( 72 , \text{cm}^3 \).