To find the volumes of Michelle's cube and Tasha's square-based pyramid, we will follow the steps below.
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 a side of the cube.
Since we know the height of the cube (which is also the side length, \( s \)), we have:
\[ s = 6 \text{ cm} \]
Now, substituting the value of \( s \) into the volume formula:
\[ V = 6^3 \] \[ V = 6 \times 6 \times 6 \] \[ V = 36 \times 6 \] \[ V = 216 \text{ cubic centimeters} \]
So, the volume of Michelle’s cube is \( 216 , \text{cm}^3 \).
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 base area and \( h \) is the height of the pyramid.
Finding the base area \( B \)
Since Tasha's pyramid has the same base area as Michelle's cube, the base area of the cube is given by the area of one of its faces, which is a square:
\[ B = s^2 = 6^2 = 36 \text{ square centimeters} \]
Finding the height \( h \)
Tasha's pyramid has the same height as Michelle’s cube:
\[ h = 6 \text{ cm} \]
Now we can substitute \( B \) and \( h \) into the volume formula for the pyramid:
\[ V = \frac{1}{3} \times 36 \times 6 \]
Calculating this step-by-step:
\[ V = \frac{1}{3} \times 216 \] \[ V = 72 \text{ cubic centimeters} \]
So, the volume of Tasha’s pyramid is \( 72 , \text{cm}^3 \).
Summary of Results
- Volume of Michelle’s cube: \( 216 , \text{cm}^3 \)
- Volume of Tasha’s pyramid: \( 72 , \text{cm}^3 \)
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