# A cone has a height of #36 cm# and its base has a radius of #9 cm#. If the cone is horizontally cut into two segments #12 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 12 cm from base, So upper radius of the frustum

Total surface area of bottom segment

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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 cone. The slant height can be found using the Pythagorean theorem, where the height (h) is one leg of the right triangle, the radius (r) is the other leg, and the slant height (l) is the hypotenuse.

Using the Pythagorean theorem: l^2 = r^2 + h^2

Plugging in the values: l^2 = 9^2 + 36^2 l^2 = 81 + 1296 l^2 = 1377 l = √1377 l ≈ 37.12 cm

Now, we have the slant height (l) of the cone. To find the surface area of the bottom segment, we need to find the area of the circular base and the lateral surface area of the cone up to the cutting point.

The area of the circular base is given by the formula: A = πr^2 A = π(9)^2 A = π(81) A = 81π square cm

The lateral surface area of the cone up to the cutting point can be calculated using the formula: A = πrl A = π(9)(37.12) A = 333.48π square cm

Now, we need to subtract the lateral surface area of the smaller cone (the top segment) from the total lateral surface area to find the lateral surface area of the bottom segment.

The total lateral surface area is 333.48π square cm, and the height of the top segment is 36 - 12 = 24 cm. So, the slant height of the top segment can be found similarly to the slant height of the whole cone.

Using the Pythagorean theorem again: l^2 = 9^2 + 24^2 l^2 = 81 + 576 l^2 = 657 l = √657 l ≈ 25.63 cm

Now, we can find the lateral surface area of the top segment: A = π(9)(25.63) A ≈ 724.39π square cm

Finally, we subtract the lateral surface area of the top segment from the total lateral surface area to find the lateral surface area of the bottom segment: 333.48π - 724.39π ≈ -390.91π square cm

Therefore, the surface area of the bottom segment of the cone is approximately 390.91π square cm.

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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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