Pearson Geometry Common Core, 2011
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Pearson Geometry Common Core, 2011 View details
6. Surface Areas and Volumes of Spheres
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Exercise 40 Page 738

Use the formula for the surface area of a sphere and the formula for the volume.

343/6Ď€m^3

Practice makes perfect

We are told that the surface area of a sphere is 49Ď€ m^2 and want to find its volume.

We first need to find its radius. To do so, we will use the formula for the surface area of a sphere. The surface area of a sphere is four times the product of π and the square of the radius. S.A.=4π r^2Let's substitute 49π for S.A. in the formula and solve for r.
S.A.=4Ď€ r^2
49Ď€=4Ď€ r^2
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Solve for r
49/4=r^2
r^2=49/4
r=sqrt(49/4)
r=sqrt(49)/sqrt(4)
r=7/2
Note that, when solving the above equation, we kept the principal root. This is because r is the radius of a sphere and therefore must be positive. The radius of a sphere with surface area 49Ď€ m^2 is 72m.
Now, let's find the volume of the sphere. The volume of a sphere is four thirds the product of π and the cube of the radius. V=4/3π r^3 Since we already know that the radius is 72, we can substitute its value for r into this formula.
V=4/3Ď€ r^3
V=4/3Ď€ ( 7/2)^3
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Simplify right-hand side
V=4/3Ď€ (7^3/2^3)
V=4/3Ď€ (343/8)
V=4/3(343/8)Ď€
V=1372/24Ď€
V=343/6Ď€
The volume of a sphere in terms of π with surface area 49π m^2 is 3436π m^3.