SOLUTION: The slant height of a cone is 2 cm more than its radius r cm.
a) Write down the expression for the total surface area of the cone, in terms of r.
b)Find r given the cone has
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-> SOLUTION: The slant height of a cone is 2 cm more than its radius r cm.
a) Write down the expression for the total surface area of the cone, in terms of r.
b)Find r given the cone has
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Question 1150803: The slant height of a cone is 2 cm more than its radius r cm.
a) Write down the expression for the total surface area of the cone, in terms of r.
b)Find r given the cone has surface area 220 pie cm^2.
c)Find the volume of the cone. Answer by jim_thompson5910(35256) (Show Source):
s = slant height
r = radius
s = r+2
since the slant height is 2 more than the radius
B = area of circular base
B = pi*r^2
C = area of curved conical surface (aka everything but the base)
C = pi*r*s
C = pi*r*(r+2)
C = pi*r*r+pi*r*2
C = pi*r^2+2pi*r
A = total surface area
A = B+C
A = pi*r^2 + pi*r*s
A = pi*r^2 + pi*r^2+2pi*r
A = 2pi*r^2+2pi*r
A = 2pi*r(r+1)
Answer: 2pi*r^2+2pi*r or 2pi*r(r+1) (both are equivalent forms)
The units for the surface area are in square cm or cm^2.
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Part (b)
A = total surface area of cone
A = 220pi .... given in instructions
A = 2pi*r(r+1) ... found back in part (a)
220pi = 2pi*r(r+1)
220 = 2*r(r+1) .... divide both sides by pi, those terms cancel
110 = r(r+1) ... divide both sides by 2
110 = r^2+r
0 = r^2+r-110
r^2+r-110 = 0
(r+11)(r-10) = 0
r+11 = 0 or r-10 = 0
r = -11 or r = 10
Ignore the negative solution as a negative radius makes no sense.
The only practical solution is r = 10.
Answer: The radius is 10 cm.
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Part (c)
h = height of cone
h = sqrt(s^2 - r^2) ... from the pythagorean theorem
h = sqrt((r+2)^2 - r^2) ... plug in s = r+2
h = sqrt((r^2+4r+4) - r^2)
h = sqrt(4r+4)
h = sqrt(4(r+1))
h = sqrt(4)*sqrt(r+1)
h = 2*sqrt(r+1)
V = volume of cone
V = (1/3)*pi*r^2*h
V = (1/3)*pi*r^2*2*sqrt(r+1)
V = (2/3)*pi*r^2*sqrt(r+1)
V = (2/3)*pi*10^2*sqrt(10+1) ... plug in r = 10 (found back in part (b))
V = (2/3)*pi*100*sqrt(11)
V = (200/3)*pi*sqrt(11)
V = 694.63227173962
Round the approximate result however you need to.
The units for the volume are in cubic cm, which can be shortened to cm^3.
The answer you end up going with (either exact or approximate) will depend on your teacher.
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