If the volume of a sphere doubles, what is the ratio of the surface area of the new, larger sphere to the old?

Answer 1

The ratio of the surface area of the new, larger sphere to the old is
#root(3)(4)#

Let's start with two formulas - for surface area of a sphere #S# and for its volume #V#, assuming the radius of a sphere is #R#: #S=4piR^2# #V=4/3piR^3#

To double the volume, we have to increase the radius by multiplying it by #root(3)(2)#. Indeed, let #R_1 = Rroot(3)(2)# Then the volume of a sphere with radius #R_1# will be #V_1 = 4/3piR_1^3 = 4/3 pi (Rroot(3)(2))^3 = 8/3piR^3# - twice larger than original volume.

With radius #R_1 = Rroot(3)(2)# the surface area of a new sphere will be #4 pi R_1^2 = 4 pi R^2(root(3)(2))^2=4 root(3)(4) pi R^2#

The ratio of the new surface area to the old one equals to #(4 root(3)(4) pi R^2) / (4 pi R^2) = root(3)(4)#

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Answer 2

Wyatt Anderson

When the volume of a sphere doubles, the ratio of the surface area of the new, larger sphere to the old sphere is ( \sqrt[3]{2} : 1 ) or approximately ( 1.26 : 1 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.