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

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

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Answer 1 · 2017-05-23T11:00:29

The surface area of the bottom segment $= 256.259 {cm}^{2}$ $= 256.259" cm"^2$

Explanation:

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

enter image source here

$Surface area of a cone = π r^{2} + π r s$ $"Surface area of a cone" = pi r^2 + pi r s$

$SA = π {DG}^{2} + π r DG \times DE$ $"SA" = pi "DG"^2 + pi r"DG" xx"DE"$

We can find out the length of line $D G$ $DG$ because since lines $C E$ $CE$ and $B E$ $BE$ are of constant gradient, $\frac{CH}{CE} = \frac{DG}{DE}$ $"CH"/"CE" = "DG"/"DE"$

$\frac{CH}{CE} = \frac{DG}{DE}$ $"CH"/"CE" = "DG"/"DE"$

$\frac{4.5}{CE} = \frac{DG}{DE}$ $4.5/"CE" = "DG"/"DE"$

${CE}^{2} = {4.5}^{2} + 36^{2}$ $"CE"^2 = 4.5^2 + 36^2$ (Pythagoras theorem)

${CE}^{2} = 20.25 + 1296$ $"CE"^2 = 20.25 + 1296$

$CE = \sqrt{1316.25}$ $"CE" = sqrt(1316.25)$

$CE = 36.28$ $"CE" = 36.28$

$\frac{4.5}{36.28} = \frac{DG}{DE}$ $4.5/"36.28" = "DG"/"DE"$

$\frac{HE}{CE} = \frac{GE}{DE}$ $"HE"/"CE" = "GE"/"DE"$

$\frac{36}{36.28} = \frac{24}{DE}$ $"36"/"36.28" = "24"/"DE"$

$\frac{36}{36.28} = \frac{24}{24.19}$ $"36"/"36.28" = "24"/"24.19"$

now we know that:

$DE = 24.19$ $"DE" = 24.19$

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

${DG}^{2} = {DE}^{2} - {GE}^{2}$ $"DG"^2 = "DE"^2 - "GE"^2$

${DG}^{2} = {24.19}^{2} - 24^{2}$ $"DG"^2 = 24.19^2 - 24^2$

$DG = \sqrt{585 - 576}$ $"DG" = sqrt(585 - 576$

$DG = \sqrt{9}$ $"DG" = sqrt(9$

$DG = 3$ $"DG" =3$

We can also do the same equation as above to find out what $DG$ $"DG"$ 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.

$SA = π 3^{2} + π 3 \times 24.19$ $"SA" = pi 3^2 + pi 3 xx 24.19$

$SA = 28.274 + 9.425 \times 24.19$ $"SA" = 28.274 + 9.425 xx 24.19$

$SA = 28.274 + 227.985$ $"SA" = 28.274 + 227.985$

$SA = 256.259 {cm}^{2}$ $"SA" = 256.259" cm"^2$

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

1 Answer

Explanation:

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A cone has a height of 36cm36 cm and its base has a radius of 9cm9 cm. If the cone is horizontally cut into two segments 24cm24 cm from the base, what would the surface area of the bottom segment be?

1 Answer

Explanation:

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