A cone has a height of #36 cm# and its base has a radius of #9 cm#. If the cone is horizontally cut into two segments #24 cm# from the base, what would the surface area of the bottom segment be?

Answer 1

The surface area of the bottom segment #= 256.259" cm"^2#

To explain this, I will show it with a diagram of a triangle as if the cone was vertically split.

#"Surface area of a cone" = pi r^2 + pi r s#

#"SA" = pi "DG"^2 + pi r"DG" xx"DE"#

We can find out the length of line #DG# because since lines #CE# and #BE# are of constant gradient, #"CH"/"CE" = "DG"/"DE"#

#"CH"/"CE" = "DG"/"DE"#

#4.5/"CE" = "DG"/"DE"#

#"CE"^2 = 4.5^2 + 36^2# (Pythagoras theorem)

#"CE"^2 = 20.25 + 1296#

#"CE" = sqrt(1316.25)#

#"CE" = 36.28#

#4.5/"36.28" = "DG"/"DE"#

#"HE"/"CE" = "GE"/"DE"#

#"36"/"36.28" = "24"/"DE"#

#"36"/"36.28" = "24"/"24.19"#

now we know that:

#"DE" = 24.19#

So we can now use the pythagorean theorem again to find out what #"DG"# is.

#"DG"^2 = "DE"^2 - "GE"^2#

#"DG"^2 = 24.19^2 - 24^2#

#"DG" = sqrt(585 - 576#

#"DG" = sqrt(9#

#"DG" =3#

We can also do the same equation as above to find out what #"DG"# is, but this was what first came to my mind when answering this question

Now that we have those measurements, we can substitute them in the Surface area equation.

#"SA" = pi 3^2 + pi 3 xx 24.19#

#"SA" = 28.274 + 9.425 xx 24.19#

#"SA" = 28.274 + 227.985#

#"SA" = 256.259" cm"^2#

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Answer 2

To find the surface area of the bottom segment of the cone after it has been horizontally cut 24 cm from the base, we need to calculate the surface area of the larger cone and subtract the surface area of the smaller cone.

First, we find the slant height of the larger cone using the Pythagorean theorem: [ l = \sqrt{h^2 + r^2} = \sqrt{36^2 + 9^2} = \sqrt{1296 + 81} = \sqrt{1377} ]

Next, we calculate the surface area of the larger cone: [ A_1 = \pi r_1 l_1 = \pi \times 9 \times \sqrt{1377} ]

Then, we find the slant height of the smaller cone (segment): [ l_2 = \sqrt{(h - 24)^2 + r^2} = \sqrt{(36 - 24)^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 ]

Now, we calculate the surface area of the smaller cone (segment): [ A_2 = \pi r_2 l_2 = \pi \times 9 \times 15 ]

Finally, we subtract the surface area of the smaller cone from the surface area of the larger cone to find the surface area of the bottom segment: [ A_{\text{bottom segment}} = A_1 - A_2 ]

[ A_{\text{bottom segment}} = \pi \times 9 \times \sqrt{1377} - \pi \times 9 \times 15 ]

[ A_{\text{bottom segment}} = \pi \times 9 \times (\sqrt{1377} - 15) ]

This gives us the surface area of the bottom segment of the cone.

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Answer from HIX Tutor

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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