To determine which design would cost less in packaging, we need to compare the surface areas of both the rectangular prism and the sphere. The surface area gives us an idea of the amount of material needed for packaging.
### Rectangular Prism
The surface area \(A\) of a rectangular prism with dimensions \(l \times w \times h\) is given by:
\[ A = 2lw + 2lh + 2wh \]
Given dimensions:
- \( l = 8 \) inches
- \( w = 1.5 \) inches
- \( h = 11 \) inches
Substitute these values into the formula:
\[ A = 2(8 \times 1.5) + 2(8 \times 11) + 2(1.5 \times 11) \]
\[ A = 2(12) + 2(88) + 2(16.5) \]
\[ A = 24 + 176 + 33 \]
\[ A = 233 \text{ square inches} \]
### Sphere
The surface area \(A\) of a sphere with radius \(r\) is given by:
\[ A = 4\pi r^2 \]
Given radius:
- \( r = 3.2 \) inches
Substitute this value into the formula:
\[ A = 4\pi (3.2)^2 \]
\[ A = 4\pi (10.24) \]
\[ A = 40.96\pi \]
Using the approximation \(\pi \approx 3.14\):
\[ A = 40.96 \times 3.14 \]
\[ A = 128.61 \text{ square inches} \]
### Conclusion
- The surface area of the rectangular prism is 233 square inches.
- The surface area of the sphere is approximately 128.61 square inches.
Since the sphere has a smaller surface area, it would require less material for packaging. Thus, the sphere with a radius of 3.2 inches would cost less in packaging compared to the rectangular prism with dimensions of 8 in. x 1.5 in. x 11 in.
5. Using your answers from above, which design would cost less in packaging?
*
the rectangular prism with dimensions of 8 in. x 1.5 in. x 11 in
the sphere with a radius of 3.2 in.
1 answer