To find the volume of a rectangular pyramid, we use the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
The base area \(A\) of a rectangle is calculated as:
\[ A = \text{Length} \times \text{Width} \]
Given the dimensions of the original pyramid are:
- Length = 24
- Width = 33
- Height = 28
First, let's calculate the base area:
\[ A = 24 \times 33 = 792 \]
Now, we can calculate the volume of the original pyramid:
\[ V = \frac{1}{3} \times 792 \times 28 \]
Calculating \(792 \times 28\):
\[ 792 \times 28 = 22176 \]
Now, we find the volume:
\[ V = \frac{1}{3} \times 22176 \approx 7392 \]
So, the volume of the original pyramid is \(7392\) cubic units.
Now that we know the volume of the original pyramid, we can find the volume of the new pyramid using the scale factor of \( \frac{2}{3} \).
The volume of similar shapes scales by the cube of the scale factor. Thus, the volume of the new pyramid \(V'\) can be calculated as follows:
\[ V' = V \times \left(\frac{2}{3}\right)^3 \]
Calculating \( \left(\frac{2}{3}\right)^3 \):
\[ \left(\frac{2}{3}\right)^3 = \frac{8}{27} \]
Now, substituting the volume of the original pyramid:
\[ V' = 7392 \times \frac{8}{27} \]
Calculating \(7392 \times 8\):
\[ 7392 \times 8 = 59136 \]
Now, dividing by 27:
\[ V' = \frac{59136}{27} = 2192 \]
So, the volume of the new pyramid is \(\boxed{2192}\) cubic units.
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