# A cone has a height of #12 cm# and its base has a radius of #2 cm#. If the cone is horizontally cut into two segments #7 cm# from the base, what would the surface area of the bottom segment be?

If we start with a cone of height 12cm and cut the top off, the height of the top section will be:

The radius of the top section (which is still a cone ) will also be reduced by the same factor:

We will call these:

The formula for the lateral surface area of a cone is given as:

Note:

The lateral surface area is the area of the cone minus the base area of the cone.

If we now find the lateral surface area of the given cone and subtract the lateral surface area of the top cone, we will be left with the lateral surface area of the bottom section. Then we can add the surface area of the base and the top of this section( this will have the same radius as the top cone section):

We can combine this into one formula:

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To find the surface area of the bottom segment of the cone, we can use the formula for the lateral surface area of a cone and subtract the area of the truncated portion.

First, let's find the slant height of the cone. Using the Pythagorean theorem in the right triangle formed by the height, radius, and slant height:

[ \text{Slant height} = \sqrt{(\text{Height})^2 + (\text{Radius})^2} ] [ \text{Slant height} = \sqrt{(12 , \text{cm})^2 + (2 , \text{cm})^2} ] [ \text{Slant height} = \sqrt{144 + 4} ] [ \text{Slant height} = \sqrt{148} ] [ \text{Slant height} = 2\sqrt{37} , \text{cm} ]

Next, we calculate the lateral surface area of the whole cone:

[ \text{Lateral surface area} = \pi \times \text{Radius} \times \text{Slant height} ] [ \text{Lateral surface area} = \pi \times 2 , \text{cm} \times 2\sqrt{37} , \text{cm} ] [ \text{Lateral surface area} = 4\pi\sqrt{37} , \text{cm}^2 ]

Now, we find the area of the truncated portion by first finding the radius of the smaller cone formed after cutting 7 cm from the base. The new height is ( 12 , \text{cm} - 7 , \text{cm} = 5 , \text{cm} ). Using similar triangles, we have:

[ \frac{\text{New radius}}{\text{Original radius}} = \frac{\text{New height}}{\text{Original height}} ] [ \frac{\text{New radius}}{2 , \text{cm}} = \frac{5 , \text{cm}}{12 , \text{cm}} ] [ \text{New radius} = \frac{5}{6} \times 2 , \text{cm} ] [ \text{New radius} = \frac{5}{3} , \text{cm} ]

Now, we can find the area of the truncated portion:

[ \text{Area of truncated portion} = \pi \times (\text{Original radius} + \text{New radius}) \times \text{Slant height} ] [ \text{Area of truncated portion} = \pi \times (2 , \text{cm} + \frac{5}{3} , \text{cm}) \times 2\sqrt{37} , \text{cm} ] [ \text{Area of truncated portion} = \pi \times \frac{11}{3} , \text{cm} \times 2\sqrt{37} , \text{cm} ] [ \text{Area of truncated portion} = \frac{22}{3}\pi\sqrt{37} , \text{cm}^2 ]

Finally, the surface area of the bottom segment (after subtracting the truncated portion) is:

[ \text{Surface area of bottom segment} = \text{Lateral surface area} - \text{Area of truncated portion} ] [ \text{Surface area of bottom segment} = 4\pi\sqrt{37} , \text{cm}^2 - \frac{22}{3}\pi\sqrt{37} , \text{cm}^2 ] [ \text{Surface area of bottom segment} = \frac{12}{3}\pi\sqrt{37} , \text{cm}^2 - \frac{22}{3}\pi\sqrt{37} , \text{cm}^2 ] [ \text{Surface area of bottom segment} = \frac{-10}{3}\pi\sqrt{37} , \text{cm}^2 ]

So, the surface area of the bottom segment of the cone is ( \frac{-10}{3}\pi\sqrt{37} , \text{cm}^2 ). Note that the negative sign indicates that the truncated portion's area is greater than the lateral surface area of the whole cone.

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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 bottom segment using the Pythagorean theorem. Then, we can use the formula for the lateral surface area of a cone to find the surface area of the bottom segment.

The slant height (l) of the bottom segment can be found using the Pythagorean theorem:

l = √(r^2 + h^2)

where r is the radius of the cone's base and h is the height of the cone.

Given that the radius (r) is 2 cm and the height (h) is 12 cm, we can calculate the slant height (l):

l = √(2^2 + 12^2) = √(4 + 144) = √148 ≈ 12.166 cm

Now, we can use the formula for the lateral surface area (A) of a cone:

A = π * r * l

Plugging in the values we have:

A = π * 2 * 12.166 ≈ 76.43 cm²

Therefore, the surface area of the bottom segment of the cone is approximately 76.43 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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