What is the surface area of the cone? Use 3.14 for pi and round to the nearest tenth, if necessary.

(radius 6) ( hight 11)

3 answers

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

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

where:

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

First, we need to calculate the slant height \( l \) using the Pythagorean theorem:

\[ l = \sqrt{r^2 + h^2} \]

Substituting the given values \( r = 6 \) and \( h = 11 \):

\[ l = \sqrt{6^2 + 11^2} = \sqrt{36 + 121} = \sqrt{157} \]

Now, calculating \( \sqrt{157} \):

\[ \sqrt{157} \approx 12.53 \]

Now that we have \( l \), we can calculate the surface area:

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

Substituting the values:

\[ \text{Surface Area} = \pi (6^2) + \pi (6)(12.53) \]

Calculating \( \pi (6^2) \):

\[ \pi (6^2) = \pi (36) \approx 3.14 \times 36 \approx 113.04 \]

Now calculating \( \pi (6)(12.53) \):

\[ \pi (6)(12.53) \approx 3.14 \times 6 \times 12.53 \approx 3.14 \times 75.18 \approx 236.60 \]

Now, adding both parts together:

\[ \text{Surface Area} \approx 113.04 + 236.60 \approx 349.64 \]

Finally, rounding to the nearest tenth:

\[ \text{Surface Area} \approx 349.6 \]

Thus, the surface area of the cone is approximately 349.6 square units.

sorry lenght is 11

No problem! Let's recalculate the surface area of the cone with the correct height of 11 units. The radius remains 6 units.

The formula for the surface area of a cone is:

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

where:

  • \( r \) = radius of the base of the cone,
  • \( h \) = height of the cone,
  • \( l \) = slant height of the cone, calculated using:

\[ l = \sqrt{r^2 + h^2} \]

Given:

  • \( r = 6 \)
  • \( h = 11 \)

Step 1: Calculate the Slant Height \( l \)

\[ l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 11^2} = \sqrt{36 + 121} = \sqrt{157} \]

Calculating \( \sqrt{157} \):

\[ \sqrt{157} \approx 12.53 \]

Step 2: Calculate the Surface Area

Now we can substitute back into the surface area formula:

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

Calculating \( \pi r^2 \) first:

\[ \pi r^2 = \pi (6^2) = \pi (36) \approx 3.14 \times 36 \approx 113.04 \]

Next, calculate \( \pi r l \):

\[ \pi r l = \pi (6)(12.53) \approx 3.14 \times 6 \times 12.53 \approx 3.14 \times 75.18 \approx 236.60 \]

Step 3: Add Both Parts Together

Now, we can add these two components together:

\[ \text{Surface Area} \approx 113.04 + 236.60 \approx 349.64 \]

Final Step: Round to the Nearest Tenth

Rounding gives us:

\[ \text{Surface Area} \approx 349.6 \]

Thus, the surface area of the cone with a radius of 6 and a height of 11 is approximately 349.6 square units.