# A cone has a height of #24 cm# and its base has a radius of #6 cm#. If the cone is horizontally cut into two segments #5 cm# from the base, what would the surface area of the bottom segment be?

Total surface area of bottom segment is 358.04 (2dp)# sq.cm.

The cone is cut at 5 cm from base, So upper radius of the frustum of

Total surface area of bottom segment

sq.cm [Ans]

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To find the surface area of the bottom segment of the cone, we first need to calculate the slant height of the bottom segment using the Pythagorean theorem. Let's denote the slant height of the bottom segment as ( l ), the height of the bottom segment as ( h ), and the radius of the cone's base as ( r ).

Given:

- Height of the cone (( H )) = 24 cm
- Radius of the base (( r )) = 6 cm
- Distance from the base where the cone is cut (( h )) = 5 cm

Using the Pythagorean theorem: [ l^2 = r^2 + (H - h)^2 ]

Substituting the given values: [ l^2 = 6^2 + (24 - 5)^2 ] [ l^2 = 36 + 19^2 ] [ l^2 = 36 + 361 ] [ l^2 = 397 ]

[ l = \sqrt{397} ]

Now, we can find the surface area (( A )) of the bottom segment using the formula: [ A = \pi r^2 + \pi r l ]

Substituting the given values: [ A = \pi (6)^2 + \pi (6)(\sqrt{397}) ] [ A = 36\pi + 6\pi\sqrt{397} ]

Thus, the surface area of the bottom segment of the cone is ( 36\pi + 6\pi\sqrt{397} ) square centimeters.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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