A cone has a height of #32 cm# and its base has a radius of #18 cm#. If the cone is horizontally cut into two segments #15 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 after it's horizontally cut 15 cm from the base, you need to first find the radius and slant height of the smaller cone formed by the bottom segment.
Using similar triangles, the radius of the smaller cone can be calculated as ( r' = \frac{h_1}{h} \times r ), where ( h_1 ) is the distance from the cut to the base (15 cm), ( h ) is the total height of the cone (32 cm), and ( r ) is the radius of the original cone (18 cm).
[ r' = \frac{15}{32} \times 18 = \frac{135}{8} \text{ cm} ]
The slant height (( l' )) of the smaller cone can be found using the Pythagorean theorem:
[ l' = \sqrt{r'^2 + h_1^2} ]
[ l' = \sqrt{\left(\frac{135}{8}\right)^2 + 15^2} = \sqrt{\frac{18225}{64} + 225} = \sqrt{\frac{18225 + 14400}{64}} = \sqrt{\frac{32625}{64}} = \frac{\sqrt{32625}}{8} \text{ cm} ]
Now, to find the surface area of the bottom segment, you calculate the lateral surface area of the smaller cone using the formula ( A = \pi r' l' ):
[ A = \pi \times \frac{135}{8} \times \frac{\sqrt{32625}}{8} ]
[ A \approx 716.49 \text{ cm}^2 ]
Therefore, the surface area of the bottom segment of the cone is approximately 716.49 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.
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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