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Question 339398: A cone of height 24cm has a curved surface area 550 cm2 . Find its volume
Answer by nyc_function(2741)   (Show Source):
You can put this solution on YOUR website! Surface area = pi * r^2 + pi * r * sqrt(r^2 + h^2)
However, we're only finding the area of pi * r * sqrt(r^2 + h^2)
550 = pi * r * sqrt(r^2 + 24^2)
550 / pi = r * sqrt(r^2 + 576)
(550 / pi)^2 = r^2 * (r^2 + 576)
0 = r^4 + 576 * r^2 - (550 / pi)^2
t = r^2
0 = t^2 + 576t - (550 / pi)^2
t = (-576 +/- sqrt(576^2 + 4 * (550 / pi)^2)) / 2
We want a positive answer
t = (-576 + 2 * sqrt(82944 + (550 / pi)^2)) / 2
t = -288 + sqrt(82944 * pi^2 + 550^2) / pi
t = -288 + (1/pi) * 1058.8316520788164279747069707806
t = 49.036582661003100584565681713644
r^2 = 49.036582661003100584565681713644
r = 7.0026125596810723871080457785158
For all intents and purposes, I'm going to presume that the radius of the cone is 7cm. From there, the volume is easy to derive. The volume of ALL right pyramids is:
(1/3) * B * h
where B = area of the base
(1/3) * B * h
B = pi * r^2
B = pi * 49
(1/3) * pi * 49 * 24
pi * 8 * 49
pi * 392
392pi cm^3
1231.5043202071989494773562062456 cm^3
It can be rounded to 1232 (cm)^3 as the volume of your cone.
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