A designer is creating a modern art sculpture of an hourglass out of steel, to be on display at the town library. The hourglass is made by two cones meeting at the top point. The designer wants to know how many square feet of steel is needed for the outside of the sculpture, if the slant height of each cone is 5 feet and the diameter is 8 feet. Use 3.14 for pi.(1 point)

Responses

326.56 square feet
326.56 square feet

226.08 square feet
226.08 square feet

653.12 square feet
653.12 square feet

113.04 square feet

1 answer

To find the amount of steel needed for the outside of the hourglass sculpture made of two cones, we need to calculate the 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:

\[ A = \pi r l \]

where:

  • \( r \) is the radius of the base of the cone
  • \( l \) is the slant height of the cone

Given:

  • The diameter of the base of the cone is 8 feet, so the radius \( r \) is:

\[ r = \frac{diameter}{2} = \frac{8}{2} = 4 \text{ feet} \]

  • The slant height \( l \) is 5 feet.

Now we can calculate the lateral surface area of one cone:

\[ A = \pi r l = 3.14 \times 4 \times 5 \]

Calculating this:

\[ A = 3.14 \times 4 = 12.56 \] \[ A = 12.56 \times 5 = 62.8 \text{ square feet} \]

This is the surface area of one cone. Since there are two cones in the hourglass sculpture:

\[ \text{Total surface area} = 2 \times 62.8 = 125.6 \text{ square feet} \]

It seems that I made a mistake in the options provided; none of them is 125.6 square feet. Let’s calculate again since we might be missing the base areas if they're specified to be included in the total calculations.

If the cones are open on the base and the bases are not part of the visible surface area, then the answer is indeed 125.6 square feet, but if they wish for the complete outer surfaces including the bases, then we need to add the area of the bases.

The area of the base of each cone (which is circular) is given by:

\[ \text{Area of base} = \pi r^2 = 3.14 \times (4^2) = 3.14 \times 16 = 50.24 \text{ square feet} \]

Total area for both bases:

\[ \text{Area of two bases} = 2 \times 50.24 = 100.48 \text{ square feet} \]

Adding the surface areas:

\[ \text{Total surface area with bases} = 125.6 + 100.48 = 226.08 \text{ square feet} \]

Thus, the amount of steel needed for the outside of the hourglass sculpture is 226.08 square feet. Therefore, the correct response is:

226.08 square feet.