# A cone has a height of #12 cm# and its base has a radius of #8 cm#. If the cone is horizontally cut into two segments #8 cm# from the base, what would the surface area of the bottom segment be?

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Given: cone of

The new shape is called a conical frustum . It has the surface area formula :

Use the cross-product:

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To find the surface area of the bottom segment of the cone, you can use the formula for the lateral surface area of a cone and subtract the area of the smaller circular base.

Given: Height of the cone, ( h = 12 ) cm Radius of the cone's base, ( r = 8 ) cm Distance from the base to the cut, ( d = 8 ) cm

First, we need to find the radius ( R ) of the circular base of the bottom segment after the cone is cut.

Using similar triangles, the ratio of the radius ( R ) of the bottom segment to the radius ( r ) of the cone is equal to the ratio of the distance from the base to the cut ( d ) to the height ( h ).

[ \frac{R}{r} = \frac{d}{h} ] [ \frac{R}{8} = \frac{8}{12} ] [ \frac{R}{8} = \frac{2}{3} ] [ R = 8 \times \frac{2}{3} ] [ R = \frac{16}{3} ] [ R = 5.\overline{3} ] cm

Now, let's calculate the lateral surface area ( S_l ) of the cone:

[ S_l = \pi r l ]

Where ( l ) is the slant height of the cone, which can be found using the Pythagorean theorem:

[ l = \sqrt{h^2 + r^2} ] [ l = \sqrt{12^2 + 8^2} ] [ l = \sqrt{144 + 64} ] [ l = \sqrt{208} ] [ l = \sqrt{4 \times 52} ] [ l = 4\sqrt{13} ]

Now, compute ( S_l ):

[ S_l = \pi \times 8 \times 4\sqrt{13} ] [ S_l = 32\pi\sqrt{13} ]

Next, calculate the area ( A ) of the smaller circular base:

[ A = \pi R^2 ] [ A = \pi \left(\frac{16}{3}\right)^2 ] [ A = \pi \times \frac{256}{9} ] [ A = \frac{256\pi}{9} ]

Finally, the surface area ( S ) of the bottom segment is:

[ S = S_l - A ] [ S = 32\pi\sqrt{13} - \frac{256\pi}{9} ]

[ S = \pi(32\sqrt{13} - \frac{256}{9}) ] [ S = \pi(32\sqrt{13} - 28.\overline{4}) ] [ S = \pi(32\sqrt{13} - 28.\overline{4}) ] [ S = \pi(32\sqrt{13} - 28.\overline{4}) ] [ S \approx \pi(32\sqrt{13} - 28.44) ] [ S \approx \pi(32\sqrt{13} - 28.44) ] [ S \approx 108.54\pi - 89.6\pi ] [ S \approx 18.94\pi ]

Thus, the surface area of the bottom segment of the cone is approximately ( 18.94\pi ) cm^2.

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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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