SOLUTION: the diameter of the hemisphere is equal to the diameter of the cone. if the hemispherical surface is equal to the lateral surface of the cone,find the total volume of the ice cream

Question 726091: the diameter of the hemisphere is equal to the diameter of the cone. if the hemispherical surface is equal to the lateral surface of the cone,find the total volume of the ice cream if the radius of the sphere is 12.5.. help me please.. thanks! :)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you have a cone and a half of a sphere sitting on top of the cone.

the base of the cone is a circle with a radius of 12.5

the base of the hemisphere is a circle that is the same size as the base of the cone.

The equations for volume and surface area of a sphere and a cone can be found in the following link:

http://math.about.com/od/formulas/ss/surfaceareavol.htm

for the sphere:
volume = 4/3 * pi * r^3
surface area = 4 * pi * r^2
r = radius

for the cone:
volume = 1/3 * pi * r^2 * h
area of base = pi * r^2
lateral surface area = pi * r * s
h = height
r = radius
s = slant height

you are given that the surface area of the hemisphere is the same as the surface area of the cone.

the surface area of the hemisphere is 1/2 the surface area of the sphere.
the formula for the surface area of the hemisphere is therefore equal to:

1/2 * 4 * pi * r^2 which is equal to 2 * pi * r^2.

The lateral surface area of the cone is equal to pi * r * s

since the surface area of the hemisphere is equal to the lateral surface area of the cone, you get:

2 * pi * r^2 = pi * r * s

since r is equal to 12.5, this formula becomes:

2 * pi * (12.5)^2 = pi * 12.5 * s

you can solve for s to get:

s = (2 * pi * (12.5)^2) / (pi * 12.5)

after cancelling of common terms in the numerator and denominator, this formula becomes:

s = 2 * 12.5 which is equal to 25.

the slant height of the cone is equal to 25.

now that you found the slant height of the cone, you can find the height of the cone.

the slant height of the cone and the height of the cone and the radius of the cone form a right triangle with one leg equal to 12.5 and the hypotenuse equal to 25.

the pythagorean formula of a^2 + b^2 = c^2 can be used to find the height.

let a = 12.5 and c = 25 and b = the height of the cone.

you get (12.5)^2 + b^2 = 25^2 which becomes:

156.25 + b^2 = 625

solve for b to get b = sqrt (625 - 156.25) = 21.65063509.

that's the height of the cone.

now that you know the height of the cone, you can find the volume of the cone.

the volume of the cone is equal to 1/3 * pi * r^2 * h.

this is equal to 1/3 * pi * (12.5)^2 * 21.65063509 which becomes:

volume of the cone = 3542.576883 cubic units.

volume of the hemisphere is equal to 1/2 the volume of the sphere which is equal to 1/2 * 4/3 * pi * r^3 which becomes:

volume of the hemisphere = 4090.615434 cubic units.

add the 2 volumes together and you get a total volume of 7633.192310 cubic units.

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