A cone has a height of $36 c m$ $36 cm$ and its base has a radius of $15 c m$ $15 cm$ . 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 $15 c m$ $15 cm$ . 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 · 2017-10-17T01:03:00

Total surface area of bottom segment $= 2419.0263$ $=2419.0263$ enter image source here

Explanation:

Slanting length ( $l_{1}$ $l_1$ ) of uncut cone $= \sqrt{36^{2} + 15^{2}}$ $= sqrt(36^2 + 15^2$
$l_{1} = \sqrt{1521} = 39 c m$ $l_1 = sqrt1521 = 39 cm$

Lateral surface area of uncut cone $= π r l$ $= pi r l$
$= π \cdot 15 \cdot 39 = 1837.8317 c m^{2}$ $= pi* 15 * 39 = 1837.8317 cm^2$

Slanting length of cut cone with height 12 cm is
$l_{2}$ $l_2$ , radius $r_{2}$ $r_2$
$\frac{l_{2}}{l_{1}} = \frac{h_{2}}{h_{1}} = \frac{r_{2}}{r_{1}}$ $l_2/l_1 = h_2/h_1 = r_2/r_1$
$\frac{l_{2}}{39} = \frac{12}{36} = \frac{r_{2}}{15}$ $l_2/39 = 12/36 = r_2/15$
$l_{2} = \frac{39 \cdot 12}{36} = 13 c m .$ $l_2 = (39*12)/36 = 13 cm.$
$r_{2} = \frac{15 \cdot 12}{36} = 5 c m$ $r_2 = (15*12)/36 = 5 cm$
Lateral surface area of cut cone $= π \cdot r_{2} \cdot l_{2}$ $= pi*r_2 * l_2$
$= π \cdot 5 \cdot 13 = 204.2035 c m^{2}$ $= pi* 5 * 13 = 204.2035 cm^2$

Lateral surface area of cut base $= (π \cdot r_{1} \cdot l_{1}) - (π \cdot r_{2} \cdot l_{2})$ $= (pi*r_1*l_1) - (pi*r_2*l_2)$
$= 1837.8317 - 204.2035 = 1633.6282 c m^{2}$ $= 1837.8317 - 204.2035 = 1633.6282 cm^2$ , (1)

Area of uncut cone base $= π \cdot r_{1}^{2} = π \cdot 15^{2} = 706.8583 c m^{2}$ $= pi*r_1^2 = pi*15^2 = 706.8583 cm^2$ , (2)

Area of cut cone base $= π \cdot r_{2}^{2} = π \cdot 5^{2} = 78.5398 c m^{2}$ $= pi*r_2^2 = pi*5^2 = 78.5398 cm^2$ , (3)

Total surface area of cut base $= (1) + (2) + (3)$ $= (1) + (2) + (3)$
$= 1633.6282 + 706.8583 + 78.5398 = 2419.0263 c m^{2}$ $= 1633.6282+706.8583+78.5398 = 2419.0263 cm^2$

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A cone has a height of $36 c m$ $36 cm$ and its base has a radius of $15 c m$ $15 cm$ . 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 36cm36 cm and its base has a radius of 15cm15 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 $15 c m$ $15 cm$ . 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?