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.

Question

Responses

113.04 square feet
113.04 square feet

653.12 square feet
653.12 square feet

226.08 square feet
226.08 square feet

326.56 square feet

Answer 1

To find the surface area of the hourglass sculpture (which consists of two cones), we need to calculate the lateral surface area of one cone and then multiply by two, since there are two 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
• \( l \) is the slant height of the cone

For this hourglass:

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

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

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

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

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

Calculating that:

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

Since there are two cones, we multiply this area by 2:

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

Now, let's double-check the choices provided:

However, it seems there's an issue with our calculations. Based on the available options and the expected area, let’s go through the computations again for total surface area more carefully.

Re-evaluating

1. Each cone's lateral surface area:

\[ \text{Surface area of one cone} = \pi r l = 3.14 \times 4 \times 5 = 62.8 \text{ square feet} \]

2. Total surface area for two cones:

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

Let's check if we did any arithmetic incorrectly:

Calculating One Cone:

• Lateral surface area:

\[ A = 3.14 \times 4 \times 5 = 3.14 \times 20 = 62.8 \text{ square feet (for 1 cone)} \]

Total for two cones:

• Surface area of two cones

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

After reviewing the calculations, it seems that there's a mismatch because the answer choices provided do not include 125.6 square feet. Based on these calculations, there may be more context needed, or the choices listed may refer to a specific interpretation of the sculpture’s design or calculations beyond simply the two cones.

But, if it strictly follows the calculations based directly on the given inputs, we derive 125.6 square feet, which is not one of your options. Your available responses should include an error in their values.

Hence, using the best accurate estimation from what we've derived with the given data, a correction or update may be needed on available choices.