# A cone has a height of #24 cm# and its base has a radius of #5 cm#. If the cone is horizontally cut into two segments #16 cm# from the base, what would the surface area of the bottom segment be?

The total surface area of the frustum of cone is

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To find the surface area of the bottom segment of the cone after it's horizontally cut 16 cm from the base, we need to first calculate the slant height of the bottom segment.

The slant height ((l)) of the cone's bottom segment can be found using the Pythagorean theorem, where (r) is the radius of the cone's base and (h) is the distance from the cut to the apex (which is the total height minus the distance of the cut from the base):

[ l = \sqrt{r^2 + (h - 16)^2} ]

Given that the radius (r) is 5 cm and the total height (h) is 24 cm, the slant height (l) can be calculated:

[ l = \sqrt{5^2 + (24 - 16)^2} = \sqrt{25 + 64} = \sqrt{89} ]

Once we have the slant height (l), we can calculate the surface area ((A)) of the bottom segment using the formula:

[ A = \pi r l ]

Substituting the values, we get:

[ A = \pi \times 5 \times \sqrt{89} ]

[ A \approx 5\pi \sqrt{89} ]

So, the surface area of the bottom segment of the cone is approximately (5\pi \sqrt{89}) 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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