Question #755fb

2 Answers
Nov 30, 2016

The surface area of a hemisphere having a volume of #250/3pi# is #50pi#

Explanation:

From https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-solids-intro/v/volume-of-a-sphere the Volume of a Sphere is:

#V = 4/3pir^3#

Therefore the volume of a hemisphere is 1/2 of this or:

#4/(3*2)pir^3 -> 2/3pir^3#

So to determine #r# we need to equation this formula to #250/3pi# and solve for #r#:

#2/3pir^3 = 250/3pi#

#3/pi * 2/3pir^3 = 3/pi * 250/3pi#

#cancel(3)/cancel(pi) * 2/cancel(3)cancel(pi)r^3 = cancel(3)/cancel(pi) * 250/cancel(3)cancel(pi)#

#2r^3 = 250#

#(2r^3)/2 = 250/2#

#r^3 = 125#

#r = 5#

https://www.math.hmc.edu/funfacts/ffiles/20004.2-3.shtml the surface area of a sphere is:

#A = 4pir^2#

Therefore the surface are of a hemisphere is 1/2 of this or:

#4pir^2/2 -> 2pir^2#

We know the radius of the square is #5# so we can substitute this into the formula and solve for #H#, the surface area of a hemisphere:

#H = 2pi5^2#

#H = 2pi*25#

#H = 50pi#

Nov 30, 2016

I got #75pi#

Explanation:

The area of a sphere is #4pir^2# so half of it is #4/2pir^2=2pir^2# and we need to evaluate the radius #r#;

The volume of the entire sphere is #V=4/3pir^3#. In your case the volume of the hemisphere should be half or: #V/2=2/3pir^3#
where the volume of the hemisphere is given as #250/3pi# (I am not sure if it is divided or multiplied by #pi# so I used multiplied) so we get:

#250/cancel(3)cancel(pi)=2/cancel(3)cancel(pi)r^3#

#r^3=250/2# so that #r=root(3)(250/2)=5#
We can use this value into the expression for half the surface of the sphere BUT....do we need also to consider the base of the hemisphere?
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Considering the base as well we get:

#S=2pir^2+pir^2=3pir^2=3pi(5^2)=75pi#