# A cone has a height of #24 cm# and its base has a radius of #8 cm#. If the cone is horizontally cut into two segments #3 cm# from the base, what would the surface area of the bottom segment be?

Total surface area of bottom segment is

The cone is cut at 3 cm from base, So upper radius of the frustum

Total surface area of bottom segment is

sq.cm [Ans]

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To find the surface area of the bottom segment of the cone, we first need to find the slant height of the cone, which is the distance from the tip of the cone to the base. We can use the Pythagorean theorem to find the slant height ((l)):

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

Given that the radius ((r)) of the cone's base is 8 cm and the height ((h)) of the cone is 24 cm:

[ l = \sqrt{8^2 + 24^2} ]

[ l = \sqrt{64 + 576} ]

[ l = \sqrt{640} ]

[ l \approx 25.298 ]

Now that we have the slant height of the cone, we can use it to find the surface area of the bottom segment. The surface area of a cone's bottom segment is given by the formula:

[ A = \pi r^2 ]

[ A = \pi (8)^2 ]

[ A = 64\pi ]

So, the surface area of the bottom segment of the cone is (64\pi) square centimeters.

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To find the surface area of the bottom segment of the cone, you need to calculate the surface area of the entire cone and then subtract the surface area of the top segment.

First, calculate the slant height of the cone using the Pythagorean theorem:

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

[ l = \sqrt{8^2 + 24^2} ] [ l = \sqrt{64 + 576} ] [ l = \sqrt{640} ] [ l ≈ 25.30 \text{ cm} ]

Next, find the area of the entire cone:

[ A_{\text{cone}} = \pi r l ]

[ A_{\text{cone}} = \pi \times 8 \times 25.30 ] [ A_{\text{cone}} ≈ 2010.619 \text{ cm}^2 ]

Now, calculate the radius of the smaller cone formed by the top segment:

[ r' = \frac{r}{h} \times (h - 3) ]

[ r' = \frac{8}{24} \times (24 - 3) ] [ r' = \frac{2}{3} \times 21 ] [ r' = 14 \text{ cm} ]

Next, find the area of the top segment:

[ A_{\text{top segment}} = \pi r' l' ]

[ l' = \sqrt{(l - 3)^2 + (24 - 3)^2} ] [ l' = \sqrt{(25.30 - 3)^2 + (24 - 3)^2} ] [ l' = \sqrt{22.30^2 + 21^2} ] [ l' ≈ 29.51 \text{ cm} ]

[ A_{\text{top segment}} = \pi \times 14 \times 29.51 ] [ A_{\text{top segment}} ≈ 1308.688 \text{ cm}^2 ]

Finally, calculate the surface area of the bottom segment:

[ A_{\text{bottom segment}} = A_{\text{cone}} - A_{\text{top segment}} ]

[ A_{\text{bottom segment}} ≈ 2010.619 - 1308.688 ] [ A_{\text{bottom segment}} ≈ 701.931 \text{ cm}^2 ]

So, the surface area of the bottom segment of the cone is approximately 701.931 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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