A cone has a height of #16 cm# and its base has a radius of #3 cm#. If the cone is horizontally cut into two segments #10 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 surface area of the entire cone first and then subtract the surface area of the top segment.
The surface area ( A ) of a cone is given by the formula: [ A = \pi r (r + \sqrt{h^2 + r^2}) ] Where:
- ( r ) is the radius of the base of the cone
- ( h ) is the height of the cone
Given:
- ( r = 3 ) cm (radius of the base)
- ( h = 16 ) cm (height of the cone)
We can calculate the surface area of the entire cone using the given values.
Then, to find the surface area of the top segment (after the cone has been cut), we need to find the radius of the top segment. Since the cone is cut 10 cm from the base, the height of the top segment is ( h_2 = 16 - 10 = 6 ) cm.
We can use similar triangles to find the radius of the top segment, which we'll call ( r_2 ): [ \frac{r_2}{r} = \frac{h_2}{h} ]
Once we have ( r_2 ), we can find the surface area of the top segment using the formula for the surface area of a cone and subtract it from the surface area of the entire cone to get the surface area of the bottom segment.
Let's calculate:
-
Surface area of the entire cone: [ A_{\text{cone}} = \pi \times 3 \times (3 + \sqrt{16^2 + 3^2}) ]
-
Radius of the top segment (( r_2 )): [ \frac{r_2}{3} = \frac{6}{16} ]
-
Surface area of the top segment: [ A_{\text{top segment}} = \pi \times r_2 \times (r_2 + \sqrt{6^2 + r_2^2}) ]
-
Surface area of the bottom segment: [ A_{\text{bottom segment}} = A_{\text{cone}} - A_{\text{top segment}} ]
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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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