Let's solve this problem step-by-step:
1. Calculate the volume of the rectangular prism:
The dimensions of the rectangular prism are:
- Length: 4.3 cm
- Width: 7.2 cm
- Height: 12.4 cm
The volume [tex]\( V \)[/tex] of a rectangular prism is given by:
[tex]\[
V = \text{length} \times \text{width} \times \text{height}
\][/tex]
Substituting the given dimensions:
[tex]\[
V = 4.3 \, \text{cm} \times 7.2 \, \text{cm} \times 12.4 \, \text{cm} = 383.904 \, \text{cm}^3
\][/tex]
2. Account for metal loss:
During the melting process, 10% of the metal is lost. Therefore, only 90% of the volume is available for the frustum.
So, the volume of the frustum [tex]\( V_f \)[/tex] is:
[tex]\[
V_f = 0.9 \times 383.904 \, \text{cm}^3 = 345.5136 \, \text{cm}^3
\][/tex]
3. Areas of the square ends of the frustum:
The frustum has square ends with sides of 3 cm and 8 cm. The areas of these squares are:
[tex]\[
A_1 = \left(3 \, \text{cm}\right)^2 = 9 \, \text{cm}^2
\][/tex]
[tex]\[
A_2 = \left(8 \, \text{cm}\right)^2 = 64 \, \text{cm}^2
\][/tex]
4. Calculate the height of the frustum:
The formula for the volume [tex]\( V \)[/tex] of a frustum of a square pyramid is given by:
[tex]\[
V_f = \frac{h}{3} \left( A_1 + A_2 + \sqrt{A_1 \times A_2} \right)
\][/tex]
We need to solve for the height [tex]\( h \)[/tex]. Rearranging the formula to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{3 V_f}{A_1 + A_2 + \sqrt{A_1 \times A_2}}
\][/tex]
Substituting the known values:
[tex]\[
h = \frac{3 \times 345.5136 \, \text{cm}^3}{9 \, \text{cm}^2 + 64 \, \text{cm}^2 + \sqrt{9 \, \text{cm}^2 \times 64 \, \text{cm}^2}}
\][/tex]
[tex]\[
h = \frac{1036.5408 \, \text{cm}^3}{9 \, \text{cm}^2 + 64 \, \text{cm}^2 + \sqrt{576 \, \text{cm}^4}}
\][/tex]
[tex]\[
h = \frac{1036.5408 \, \text{cm}^3}{9 \, \text{cm}^2 + 64 \, \text{cm}^2 + 24 \, \text{cm}^2}
\][/tex]
[tex]\[
h = \frac{1036.5408 \, \text{cm}^3}{97 \, \text{cm}^2}
\][/tex]
[tex]\[
h \approx 10.685987628865979 \, \text{cm}
\][/tex]
Thus, the thickness (or height) of the frustum is approximately [tex]\( 10.686 \, \text{cm} \)[/tex].
