To calculate the surface area of the hourglass sculpture made from two cones, we first need to find the lateral surface area of one cone and then double it since there are two identical cones.
The formula for the lateral surface area \( A \) of a cone is given by:
\[ A = \pi r l \]
where \( r \) is the radius of the base of the cone and \( l \) is the slant height.
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Find the radius: The diameter of the cone is 8 feet, so the radius \( r \) is:
\[ r = \frac{diameter}{2} = \frac{8 \text{ ft}}{2} = 4 \text{ ft} \]
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Use the slant height: The slant height \( l \) is given as 5 feet.
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Calculate the lateral surface area of one cone:
\[ A = \pi r l = 3.14 \times 4 \text{ ft} \times 5 \text{ ft} \]
\[ A = 3.14 \times 20 = 62.8 \text{ square feet} \]
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Calculate the total surface area for two cones:
Since there are two cones, the total surface area \( A_{total} \) is:
\[ A_{total} = 2 \times A = 2 \times 62.8 \text{ square feet} = 125.6 \text{ square feet} \]
From the options given, it appears that there might be a misunderstanding, as the calculated total surface area does not match any of the provided responses. However, we calculated based on the assumption that the bases are not included in the surface area since they are joined together at the top point.
For the outside of the sculpture, the total amount of steel needed is:
\[ 125.6 \text{ square feet} \]
If the bases of the cones were also supposed to be included (which would often be the case in such designs), you would have to calculate the area of the circular base:
- Calculate the area of the base of one cone:
The area \( A_{base} \) of one base is given by:
\[ A_{base} = \pi r^2 = 3.14 \times (4 \text{ ft})^2 = 3.14 \times 16 = 50.24 \text{ square feet} \]
Since there are two bases (one for each cone), the total base area is:
\[ A_{bases} = 2 \times 50.24 \text{ square feet} = 100.48 \text{ square feet} \]
- Adding the base areas to the total lateral area:
If we include the area of both bases to the previously calculated total surface area:
\[ A_{total_with_bases} = 125.6 \text{ square feet} + 100.48 \text{ square feet} = 226.08 \text{ square feet} \]
Thus, if the sculpture will account for the entire outer surface including bases, the total would be 226.08 square feet.
So the correct response from the options provided is 226.08 square feet.