What is the surface area, in square centimeters, of a cone with a diameter of 12 feet and a slant height of 14 feet? Use 3.14 for pi and round to the nearest tenth, if necessary

1 answer

To calculate the surface area of a cone, we can use the formula:

\[ \text{Surface Area} = \pi r^2 + \pi r l \]

where:

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

Given:

  • The diameter of the cone is 12 feet.
  • The slant height \( l \) is 14 feet.

First, we need to find the radius \( r \) from the diameter:

\[ r = \frac{\text{diameter}}{2} = \frac{12 \text{ feet}}{2} = 6 \text{ feet} \]

Now, substituting the values into the surface area formula:

  1. Calculate the area of the base:

\[ \pi r^2 = 3.14 \times (6 \text{ feet})^2 = 3.14 \times 36 \text{ square feet} = 113.04 \text{ square feet} \]

  1. Calculate the lateral area:

\[ \pi r l = 3.14 \times 6 \text{ feet} \times 14 \text{ feet} \]

\[ = 3.14 \times 84 \text{ square feet} = 263.76 \text{ square feet} \]

  1. Adding the base area and the lateral area:

\[ \text{Surface Area} = 113.04 \text{ square feet} + 263.76 \text{ square feet} = 376.8 \text{ square feet} \]

Now, convert the surface area from square feet to square centimeters. Since 1 foot is equivalent to 30.48 centimeters, the conversion factor for square feet to square centimeters is:

\[ (30.48 \text{ cm})^2 = 929.0304 \text{ cm}^2 \]

Thus, converting square feet to square centimeters:

\[ \text{Surface Area in cm}^2 = 376.8 \text{ square feet} \times 929.0304 \text{ cm}^2/\text{square foot} \approx 350,222.5 \text{ cm}^2 \]

Finally, rounding to the nearest tenth, the surface area of the cone is:

\[ \boxed{350222.5 , \text{cm}^2} \]