To find the volume of Danny's pyramid, we need to perform the following steps:
1. Determine the scaled dimensions of Danny's pyramid, given that the scale factor is [tex]\( \frac{1}{80} \)[/tex] of the original Great Pyramid.
2. Calculate the base area of Danny's pyramid.
3. Calculate the volume of Danny's pyramid using the formula for the volume of a pyramid.
### Step-by-Step Solution
1. Determine the scaled dimensions
- The original height of the Great Pyramid is 480 feet. Scaling this by [tex]\( \frac{1}{80} \)[/tex]:
[tex]\[
\text{Height of Danny's pyramid} = 480 \times \frac{1}{80} = 6 \text{ feet}
\][/tex]
- The original side length of the base is 755 feet. Scaling this by [tex]\( \frac{1}{80} \)[/tex]:
[tex]\[
\text{Side length of the base} = 755 \times \frac{1}{80} = 9.4375 \text{ feet}
\][/tex]
2. Calculate the base area
The base area of a pyramid with a square base is given by the side length squared:
[tex]\[
\text{Base area} = 9.4375^2 = 89.06640625 \text{ square feet}
\][/tex]
3. Calculate the volume
The volume [tex]\( V \)[/tex] of a pyramid is given by the formula:
[tex]\[
V = \frac{1}{3} \times \text{Base area} \times \text{Height}
\][/tex]
Plugging in the values we calculated:
[tex]\[
V = \frac{1}{3} \times 89.06640625 \times 6 = 178.1328125 \text{ cubic feet}
\][/tex]
### Conclusion
The approximate volume of Danny's pyramid is [tex]\( 178.1 \, \text{cubic feet} \)[/tex].
Thus, the correct answer is:
[tex]\[
178.1 \, ft^3
\][/tex]
Hence, the appropriate choice from the given options is:
[tex]\[
178.1 \, ft^3
\][/tex]
