# A cone has a height of #7 cm# and its base has a radius of #5 cm#. If the cone is horizontally cut into two segments #1 cm# from the base, what would the surface area of the bottom segment be?

Total surface area of bottom segment is

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To find the surface area of the bottom segment of the cone, we need to calculate the area of the circular base and the lateral surface area of the remaining frustum (the portion of the cone above the cut).

First, let's calculate the area of the circular base using the formula for the area of a circle: ( \pi \times radius^2 ). Given that the radius ( r ) of the circular base is 5 cm, the area of the base is:

[ A_{\text{base}} = \pi \times 5^2 = 25\pi \text{ cm}^2 ]

Next, we need to find the slant height of the frustum, which can be calculated using the Pythagorean theorem. The slant height (( l )) and the height (( h )) of the frustum form a right triangle, where ( l ) is the hypotenuse and ( h ) is one of the legs. We know that the height of the frustum is 6 cm (7 cm - 1 cm = 6 cm), and the radius of the circular base is 5 cm. Thus, the slant height can be found as follows:

[ l = \sqrt{r^2 + h^2} = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61} \text{ cm} ]

Now, let's calculate the lateral surface area of the frustum using the formula ( A_{\text{lateral}} = \pi \times (r_1 + r_2) \times l ), where ( r_1 ) is the radius of the base and ( r_2 ) is the radius of the top surface of the frustum (which is 1 cm less than the radius of the base).

[ A_{\text{lateral}} = \pi \times (5 + 4) \times \sqrt{61} = 9\pi \times \sqrt{61} \text{ cm}^2 ]

Therefore, the surface area of the bottom segment (the circular base) is ( 25\pi ) 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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