To estimate the volume of the hollow pyramidal shell, we will find the volumes of the outer and inner pyramids separately and then subtract the volume of the inner pyramid from the outer pyramid.
Given:
- Height [tex]\( h = 13 \)[/tex] inches
- Outer pyramid: Base length [tex]\( L_1 = 4.2 \)[/tex] inches, Base width [tex]\( W_1 = 4.2 \)[/tex] inches
- Inner pyramid: Base length [tex]\( L_2 = 4 \)[/tex] inches, Base width [tex]\( W_2 = 4 \)[/tex] inches
The volume [tex]\( V \)[/tex] of a pyramid can be calculated using the formula:
[tex]\[ V = \frac{L \cdot W \cdot H}{3} \][/tex]
Step-by-Step Solution:
1. Calculate the volume of the outer pyramid:
[tex]\[ V_1 = \frac{L_1 \cdot W_1 \cdot h}{3} \][/tex]
[tex]\[ V_1 = \frac{4.2 \cdot 4.2 \cdot 13}{3} \][/tex]
[tex]\[ V_1 \approx 76.44 \, \text{cubic inches} \][/tex]
2. Calculate the volume of the inner pyramid:
[tex]\[ V_2 = \frac{L_2 \cdot W_2 \cdot h}{3} \][/tex]
[tex]\[ V_2 = \frac{4 \cdot 4 \cdot 13}{3} \][/tex]
[tex]\[ V_2 \approx 69.333 \, \text{cubic inches} \][/tex]
3. Calculate the volume of the shell:
[tex]\[ \Delta V = V_1 - V_2 \][/tex]
[tex]\[ \Delta V \approx 76.44 - 69.333 \][/tex]
[tex]\[ \Delta V \approx 7.107 \, \text{cubic inches} \][/tex]
Therefore, the estimated volume of the hollow shell is approximately [tex]\( 7.107 \, \text{cubic inches} \)[/tex].
