The height of a cone increases by k% its semi-vertical angle remaining the same. What is the approximate percentage increase in (1) in total surface area (2) in volume assuming that k is small.?
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semi vertical angle = t
height = h
slant height= h sec (t)
radius of base = h tan (t)
surface area=S= λh² sec(t)tan(t)
dS = 2λh sec(t) tan(t) dh
dividing these two equations we will get
dS / S = 2dh / h x 100
increase in % = 2k
now
V = (λ/3 ) (htan t)²h
V = (λ/3 ) tan²t h³
dV = λ tan²t h² dh
dV/ V x100 = 3 dh /h x100
= 3k%
height = h
slant height= h sec (t)
radius of base = h tan (t)
surface area=S= λh² sec(t)tan(t)
dS = 2λh sec(t) tan(t) dh
dividing these two equations we will get
dS / S = 2dh / h x 100
increase in % = 2k
now
V = (λ/3 ) (htan t)²h
V = (λ/3 ) tan²t h³
dV = λ tan²t h² dh
dV/ V x100 = 3 dh /h x100
= 3k%
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