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)

Question

Responses

653.12 square feet
653.12 square feet

113.04 square feet
113.04 square feet

326.56 square feet
326.56 square feet

226.08 square feet

Answer 1

To find the total surface area of the hourglass sculpture, we need to calculate the lateral surface area of both cones, as the top and bottom bases will not be included in the display.

Step 1: Calculate the lateral surface area of one cone.

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

Step 2: Find the radius.

Given the diameter is 8 feet, the radius \( r \) is:

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

Step 3: Use the slant height.

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

Step 4: Calculate the lateral surface area of one cone.

Substituting the values into the formula:

\[ A = \pi r l = 3.14 \times 4 \times 5 \] \[ A = 3.14 \times 20 = 62.8 \text{ square feet} \]

Step 5: Calculate the lateral surface area of both cones.

Since the hourglass consists of two identical cones, we multiply the area of one cone by 2:

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

However, let's recheck the calculations step by step to make sure everything is accounted for.

Recalculating the final area:

1. One cone's lateral surface area: \[ 3.14 \times 4 \times 5 = 62.8 \text{ square feet} \]
2. Two cones: \[ 62.8 \times 2 = 125.6 \text{ square feet} \]

After going through the calculations, it seems there was an error in the interpretation of the question initially in relation to the total area needed. Therefore, none of the provided options seem to represent the right calculation, as they imply a calculation method that included possibly more parts or surfaces.

The calculations strictly yield \( 125.6 \) square feet needed for both cones. Please double-check if the assumption is indeed about a straightforward cone model or if additional areas were supposed to be considered, but based on the given definitions and standard calculations above, we find no match with the available choices (653.12, 113.04, 326.56, or 226.08 square feet). Let's confirm we apply them consistently.

If the designer means adding the bases, then the calculation varies, but for just the outer parts covered, it stays at \( 125.6 \text{ ft}^2 \). Could you clarify or confirm the parts noted truly relate to this scenario?