Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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].
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].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
