A spyglass in the shape of a cone has a slant height of 18 centimeters and a radius of 6 centimeters. What is the surface area of the spyglass? Use 3.14 for pi.(1 point)

Question

Responses

452.16 square centimeters
452.16 square centimeters

197.82 square centimeters
197.82 square centimeters

1,356.48 square centimeters
1,356.48 square centimeters

339.12 square centimeters

Answer 1

To find the surface area of the cone-shaped spyglass, we need to calculate both the lateral surface area and the area of the base.

1. Lateral Surface Area of a cone formula: \[ \text{Lateral Surface Area} = \pi r l \] where:

   • \( r \) is the radius (6 cm)
   • \( l \) is the slant height (18 cm)

   Substituting the values: \[ \text{Lateral Surface Area} = 3.14 \times 6 \times 18 \] \[ = 3.14 \times 108 \] \[ = 339.12 \text{ square centimeters} \]

2. Area of the Base of the cone formula: \[ \text{Area of the Base} = \pi r^2 \] Substituting the radius: \[ \text{Area of the Base} = 3.14 \times 6^2 \] \[ = 3.14 \times 36 \] \[ = 113.04 \text{ square centimeters} \]

3. Total Surface Area of the cone: \[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Area of the Base} \] \[ = 339.12 + 113.04 \] \[ = 452.16 \text{ square centimeters} \]

Thus, the surface area of the spyglass is 452.16 square centimeters.

The correct response is: 452.16 square centimeters.

Answer 2

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

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 3

To calculate the surface area of the hourglass sculpture made of two cones, we need to find the lateral surface area of one cone and then double it, since there are two identical cones.

1. Find the radius of the cone: The diameter is given as 8 feet, so the radius \( r \) is: \[ r = \frac{\text{diameter}}{2} = \frac{8}{2} = 4 \text{ feet} \]

2. Lateral Surface Area of one cone: The formula for the lateral surface area of a cone is: \[ \text{Lateral Surface Area} = \pi r l \] where:

   • \( r = 4 \) feet is the radius.
   • \( l = 5 \) feet is the slant height.

   Substituting the values: \[ \text{Lateral Surface Area} = 3.14 \times 4 \times 5 \] \[ = 3.14 \times 20 \] \[ = 62.8 \text{ square feet} \]

3. Total Surface Area for the hourglass: Since the hourglass consists of two cones, we need to double the lateral surface area: \[ \text{Total Surface Area} = 2 \times 62.8 = 125.6 \text{ square feet} \]

Given the responses you provided, none of them match the calculated surface area of 125.6 square feet. However, if you are looking for total square feet necessary to cover only the lateral surfaces of both cones, it seems the calculation aligns with my understanding.

Please verify the problem's specifications or provided options, as the calculated surface area does not match the expected answers.

If I made a mistake in the choices, please let me know!

Answer 4

It's ok. you made a mistake the choices are
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 5

Thank you for your patience! Let's go through the calculations again to ensure we get the correct surface area of the hourglass sculpture made from two cones.

Given:

• Diameter of the cone (d) = 8 feet
• Radius (r) = \( \frac{d}{2} = \frac{8}{2} = 4 \) feet
• Slant height (l) = 5 feet

Calculation of Lateral Surface Area for One Cone:

The formula for the lateral surface area of a cone is: \[ \text{Lateral Surface Area} = \pi r l \]

Substituting the values in: \[ \text{Lateral Surface Area} = 3.14 \times 4 \times 5 \] Calculating it step-by-step: \[ 3.14 \times 4 = 12.56 \] \[ 12.56 \times 5 = 62.8 \text{ square feet} \]

Total Surface Area of the Hourglass:

Since there are two identical cones, we need to double the lateral surface area: \[ \text{Total Surface Area} = 2 \times 62.8 = 125.6 \text{ square feet} \]

Reviewing the Choices:

The closest option based on calculations and considering potential adjustments for errors is:

• 113.04 square feet
• 226.08 square feet (which is the next choice that may agree with half of the area we calculated)

However, with no matches reflecting the value of 125.6 square feet, it seems there might have been an oversight in provided options or we should multiply the lateral area by some factor depending on the requirements.

It seems that based on the original calculation I provided,

The calculated total area doesn’t exactly match the available choices. Please clarify if I should approach it differently!