A cone has a height of #8 cm# and its base has a radius of #6 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?
Area is
We can see that the area of the frustum is just the difference between the area difference of the sector 2 concentric circles.
This area difference is essentially the formula for the area of a frustum:
Ignoring Figure 2 and its labels,
Back to Figure 1, the height
Recall
In the cone, image
However, the opposite side of
Substitute
Phew!
Now, we can finally calculate the area of the frustum!
where
Now we can calculate the areas of the top and bottom circles easily and add it all together:
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To find the surface area of the bottom segment of the cone:

Calculate the slant height ((l)) of the cone using the height ((h)) and the radius ((r)) of the base using the Pythagorean theorem: [ l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 , \text{cm} ]

Determine the radius ((r_1)) of the circular base of the bottom segment. This can be found using similar triangles: [ \frac{r_1}{r} = \frac{h_1}{h} ] [ \frac{r_1}{6} = \frac{1}{8} \times (8  7) ] [ r_1 = \frac{1}{8} \times 1 \times 6 = \frac{3}{4} , \text{cm} ]

Calculate the surface area ((A_1)) of the bottom segment of the cone using the formula for the surface area of a cone: [ A_1 = \pi r_1 l = \pi \times \frac{3}{4} \times 10 = \frac{30\pi}{2} = 15\pi , \text{cm}^2 ]
So, the surface area of the bottom segment of the cone is ( 15\pi , \text{cm}^2 ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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