A spyglass in the shape of a cone has a slant height of 18 cm and a radius of 6 cm what is the surface area of the spylass use 3.14 as pi

5 answers

To find the surface area of a cone, you can use the formula:

\[ \text{Surface Area} = \pi r (r + l) \]

where:

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

In this case:

  • \( r = 6 \) cm,
  • \( l = 18 \) cm,
  • \( \pi \) is given as 3.14.

Now, substituting the values into the formula:

  1. Calculate the lateral surface area:

\[ \text{Lateral Surface Area} = \pi r l = 3.14 \times 6 \times 18 \]

Calculating this:

\[ = 3.14 \times 6 = 18.84 \] \[ = 18.84 \times 18 = 338.88 \text{ cm}^2 \]

  1. Calculate the base area:

\[ \text{Base Area} = \pi r^2 = 3.14 \times (6)^2 = 3.14 \times 36 \]

Calculating this:

\[ = 3.14 \times 36 = 113.04 \text{ cm}^2 \]

  1. Finally, add the lateral surface area and the base area to find the total surface area:

\[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Base Area} = 338.88 + 113.04 = 451.92 \text{ cm}^2 \]

Thus, the surface area of the spyglass is approximately:

\[ \text{Surface Area} \approx 451.92 \text{ cm}^2 \]

thats totally wrong-

I apologize for any confusion! Let's go through the calculations again step by step to ensure we get the correct surface area of the cone-shaped spyglass.

The formula for the surface area of a cone is:

\[ \text{Surface Area} = \pi r (r + l) \]

where:

  • \( r \) = radius of the base,
  • \( l \) = slant height.

Given:

  • \( r = 6 \) cm,
  • \( l = 18 \) cm,
  • \( \pi = 3.14 \).

Step 1: Calculate the lateral surface area

\[ \text{Lateral Surface Area} = \pi r l = 3.14 \times 6 \times 18 \]

Calculating this:

  1. Calculate \( 6 \times 18 \): \[ 6 \times 18 = 108 \]

  2. Then multiply by \( \pi \): \[ 3.14 \times 108 = 339.12 \text{ cm}^2 \]

Step 2: Calculate the base area

\[ \text{Base Area} = \pi r^2 = 3.14 \times (6^2) = 3.14 \times 36 \]

Calculating this:

\[ 3.14 \times 36 = 113.04 \text{ cm}^2 \]

Step 3: Total surface area

Now, add the lateral surface area and the base area together:

\[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Base Area} \]

\[ = 339.12 + 113.04 = 452.16 \text{ cm}^2 \]

So, the correct surface area of the spyglass is approximately:

\[ \text{Surface Area} \approx 452.16 \text{ cm}^2 \]

Thank you for your patience, and I hope this clears up any confusion!

thank you and you sure thats correct just wanna make sure

Yes, let's carefully verify the calculations one more time to ensure everything is correct:

Given values:

  • Radius \( r = 6 \) cm
  • Slant height \( l = 18 \) cm
  • \( \pi \approx 3.14 \)

Step 1: Calculate the lateral surface area

The formula for the lateral surface area of a cone is:

\[ \text{Lateral Surface Area} = \pi r l \]

Substituting the values:

\[ \text{Lateral Surface Area} = 3.14 \times 6 \times 18 \]

Calculating this step-by-step:

  1. Calculate \( 6 \times 18 = 108 \).
  2. Then multiply by \( 3.14 \): \[ 3.14 \times 108 = 339.12 \text{ cm}^2 \]

Step 2: Calculate the base area

The formula for the base area of a cone is:

\[ \text{Base Area} = \pi r^2 \]

Substituting the values:

\[ \text{Base Area} = 3.14 \times (6^2) = 3.14 \times 36 \]

Calculating this:

\[ 3.14 \times 36 = 113.04 \text{ cm}^2 \]

Step 3: Total surface area

Now we add the lateral surface area and the base area:

\[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Base Area} \] \[ \text{Total Surface Area} = 339.12 + 113.04 = 452.16 \text{ cm}^2 \]

Conclusion

Thus, the total surface area of the spyglass is:

\[ \text{Surface Area} \approx 452.16 \text{ cm}^2 \]

These calculations have been thoroughly verified, and the answer provided is indeed correct. Thank you for your patience, and if you have any additional questions, feel free to ask!