To find the volume of a pyramid, the formula is:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
First, we need to calculate the volume of the original pyramid with base dimensions \( B = 24 \) and \( L = 22 \), and a height \( H = 28 \).
- Calculate the base area:
\[ \text{Base Area} = B \times L = 24 \times 22 = 528 \]
- Calculate the volume of the original pyramid:
\[ V = \frac{1}{3} \times \text{Base Area} \times H = \frac{1}{3} \times 528 \times 28 \]
Calculating this step by step:
\[ 528 \times 28 = 14784 \] \[ V = \frac{1}{3} \times 14784 \approx 4928 \]
Now that we have the volume of the original pyramid, we can calculate the volume of the new pyramid that has a scale factor of \( \frac{2}{3} \).
The volume of similar solids scales with the cube of the scale factor:
\[ \text{Volume of New Pyramid} = V \times \left( \text{Scale Factor}^3 \right) \] \[ \text{Scale Factor}^3 = \left( \frac{2}{3} \right)^3 = \frac{8}{27} \]
Now we can calculate the volume of the new pyramid:
\[ \text{Volume of New Pyramid} = 4928 \times \frac{8}{27} \]
Calculating this step by step:
\[ 4928 \times 8 = 39424 \] \[ \frac{39424}{27} \approx 1457.56 \]
Thus, the volume of the new pyramid is approximately \( \boxed{1457.56} \).