3. A hollow pyramidal shell is made out of foamboard with a fixed height of 13 inches. The square base measures 4 inches by 4 inches on the inside and 4.2 inches by 4.2 inches on the outside. Use differentials to estimate the volume of the shell, [tex]\Delta V=V(4.2)-V(4)[/tex].

For a pyramid, [tex]V=\frac{L \cdot W \cdot H}{3}[/tex].



Answer :

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].