Question

Surface Area of Cones Practice

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Question
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A cone shows a radius of 6 and hypotenuse or side as 11.

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

Answer 1

[AI-generated answer — Bot GPT-4 Turbo]

To calculate the surface area of a cone, we need to know both the area of the base (a circle) and the lateral surface area (the area of the cone's side). The total surface area (SA) is the sum of the base area and lateral surface area.

First, let's calculate the area of the base of the cone, which is a circle.

\[ \text{Base Area} = \pi r^2 \]
where \(\pi \approx 3.14\) and \(r\) is the radius of the base.

Plugging in the given values:
\[ \text{Base Area} = 3.14 \times 6^2 \]
\[ \text{Base Area} = 3.14 \times 36 \]
\[ \text{Base Area} = 113.04 \]

Next, let's calculate the lateral surface area of the cone, a.k.a. the cone's side.

The lateral surface area (LSA) is given by the formula:
\[ \text{LSA} = \pi r l \]
where \(l\) is the slant height (side) of the cone.

Based on the information provided:
\[ \text{LSA} = 3.14 \times 6 \times 11 \]
\[ \text{LSA} = 3.14 \times 66 \]
\[ \text{LSA} = 207.24 \]

Now, let's find the total surface area:
\[ \text{SA} = \text{Base Area} + \text{LSA} \]
\[ \text{SA} = 113.04 + 207.24 \]
\[ \text{SA} = 320.28 \]

Thus, the surface area of the cone, rounded to the nearest tenth, is:
\[ \text{SA} \approx 320.3 \] square units.

Answer 2

the answer to the question is 320.3 for anyone who is still doing this

Answer 3

wrong

Answer 4

[AI-generated answer — Explain Bot]

To find the surface area of a cone, you need to know the radius and slant height of the cone. The slant height is also the hypotenuse of the triangle formed by the radius, height, and slant height of the cone.

In this case, the radius of the cone is given as 6 and the slant height (hypotenuse) is given as 11. To find the height of the cone, we can use the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

In this case, the square of the height can be found by subtracting the square of the radius from the square of the slant height:
Height^2 = Slant height^2 - Radius^2
Height^2 = 11^2 - 6^2
Height^2 = 121 - 36
Height^2 = 85

To find the height, we take the square root of both sides:
Height = √85
Height ≈ 9.2 (rounded to the nearest tenth)

Now that we know the radius (6) and the height (9.2), we can find the surface area of the cone using the formula:
Surface Area = π * radius * (radius + slant height)

Surface Area = 3.14 * 6 * (6 + 11)
Surface Area = 3.14 * 6 * 17
Surface Area ≈ 319.908 (rounded to the nearest tenth)

Therefore, the surface area of the cone is approximately 319.9 square units.