SOLUTION: Hello, I have two Similar cones Large and small. Large cone is 9cm radius 15 cm slant height. The small cone has a lateral area of 60 pi. How do I find the volume of the small

Algebra ->  Surface-area -> SOLUTION: Hello, I have two Similar cones Large and small. Large cone is 9cm radius 15 cm slant height. The small cone has a lateral area of 60 pi. How do I find the volume of the small       Log On


   

Question 1083682: Hello, I have two Similar cones Large and small. Large cone is 9cm radius 15 cm slant height. The small cone has a lateral area of 60 pi.
How do I find the volume of the small cone?
Found 2 solutions by MathLover1, addingup:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Large cone is+r%5B1%5D=9cm and L%5B1%5D=15cm+ slant height
small cone has a lateral area of LA=60+pi
The lateral area of a cone is
LA=pi+%2A+r+%2A+L where r is the radius and L is the slant height
60pi=pi+%2A+r+%2A+L
60=+r+%2A+L
60%2FL=+r+

since cones similar, radii and slant heights are proportional:
r%5B1%5D%2Fr=L%5B1%5D%2FL
9%2F%2860%2FL%29=15%2FL
9L%2F60=15%2FL
3L%2F20=15%2FL
3L%2AL=15%2A20
L%5E2=%2815%2A20%29%2F3
L%5E2=5%2A20
L%5E2=100
L=10cm-> slant height of a small cone

60%2FL=+r+
60%2F10cm=+r+
6cm=+r+-> radius of a small cone

now we can find a volume:
V=%281%2F3%29b%2Ah where b is base and h height of the cone
base is a circle: area is r%5E2pi
so, V=%281%2F3%29r%5E2%2Api%2Ah
we will find height using Pythagorean theorem:
h%5E2=L%5E2-r%5E2 ->h=sqrt%2810%5E2-6%5E2%29 ->h=sqrt%28100-36%29->h=sqrt%2864%29->h=8cm
V=%281%2F3%29%286cm%29%5E2%2Api%2A8cm
V=%281%2Fcross%283%29%29%28cross%2836%2912cm%5E2%2Api%2A8cm%29
V=12cm%5E2%2Api%2A8cm
V=96cm%5E3%2Api



Answer by addingup(3677) About Me  (Show Source):
You can put this solution on YOUR website!
radius: r
length of the slant side: l
height: h
-------------
height: h = sqrt(l^2-r^2)
your large cone:
h = sqrt(15^2-9^2) = 12
------------------
Area of the base : Pi*r^2
Lateral area . . : Pi*r*l (large cone: Pi*9*15 = Pi*135 = 424.12)
Area of the cone: Pi*r^2+Pi*r*l
your large cone:
area = (Pi*9^2)+(Pi*9*15) = 254.47+424.12 = 678.58 is the total area of the large cone
----------
Volume of the cone: (Pi*r^2*h)/3 = (Pi*81*12)/3 = 1,017.88
---------------------
Your small cone has a lateral area of 60*Pi = 188.50 and from the formula we know that 60 = r*l. But here I get stuck. How are your cones similar? Do they have the same lateral area ratio, 424.12:188.50?
I somehow don't think so, I think it's something else. Do they have the same slant height? Or the same height? or the same radius?