SOLUTION: Find the volume in terms of pi of a sphere whose surface area is {{{36*pi^2}}}. The formula for the volume of a sphere is {{{ V=(4/3)*(pi)*r^3}}} and for the surface area of a sphe

Algebra ->  Surface-area -> SOLUTION: Find the volume in terms of pi of a sphere whose surface area is {{{36*pi^2}}}. The formula for the volume of a sphere is {{{ V=(4/3)*(pi)*r^3}}} and for the surface area of a sphe      Log On


   

Question 76989This question is from textbook PRENTICE HALL MATHEMATICS GEOMETRY
: Find the volume in terms of pi of a sphere whose surface area is 36%2Api%5E2. The formula for the volume of a sphere is +V=%284%2F3%29%2A%28pi%29%2Ar%5E3 and for the surface area of a sphere is S+=+4%2Api%2Ar%5E2 This question is from textbook PRENTICE HALL MATHEMATICS GEOMETRY

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
I've taken the liberty of changing the problem a little to add the correct formula for the
Surface Area (call it S) of a sphere. This formula is S+=+4%2A%28pi%29%2Ar%5E2
.
You are told that for a particular sphere, the surface area is 36%2A%28pi%29%5E2. Substitute
this value into the surface area equation in place of S and you get:
.
36%2A%28pi%29%5E2+=+4%2A%28pi%29%2Ar%5E2
.
What we are going to do now is solve this equation for r. Begin by dividing both sides of
the equation by 4. When you do that the equation reduces to:
.
9%2A%28pi%29%5E2+=+%28pi%29%2Ar%5E2
.
Next divide both sides of the equation by %28pi%29. This further reduces the equation to:
.
9%2A%28pi%29+=+r%5E2
.
Since we're going to solve for r, let's transpose the equation (switch left and right sides)
to get it into the more conventional form:
.
r%5E2+=+9%2A%28pi%29
.
Now you can solve for r by taking the square root of both sides. When you do that you
get:
.
r+=+sqrt%289%2A%28pi%29%29+=+sqrt%289%29%2Asqrt%28pi%29+=+3%2Asqrt%28pi%29
.
We now know what the radius of the given sphere is. To find the volume (V) of this sphere
we substitute this value for the radius into the volume equation. The volume equation is:
.
V+=+%284%2F3%29%2A%28pi%29%2Ar%5E3
.
substitute 3%2Asqrt%28pi%29 for the radius in the volume equation and you get:
.
V+=+%284%2F3%29%2A%28pi%29%2A%283%2Asqrt%28pi%29%29%5E3
.
By the rules of exponents, when you cube a factored quantity, it is equivalent to cubing
each of the individual factors. Therefore, %283%2Asqrt%28pi%29%29%5E3+=+%283%5E3%29%2A+%28sqrt%28pi%29%29%5E3.
But 3%5E3+=+3%2A3%2A3+=+27 and
.
Substitute these values into the volume equation and you get:
.
V+=+%284%2F3%29%2A%28pi%29%2A27%2A%28pi%29%2Asqrt%28pi%29
.
But some of the factors on the right side can be multiplied into a simpler form. The product
of %284%2F3%29%2A27 is 36 and the product of %28pi%29%2A%28pi%29 is %28pi%29%5E2. Substituting
these into the volume equation results in:
.
V+=+36%2A%28pi%29%5E2%2Asqrt%28pi%29
.
For the given sphere whose surface area is 36%2A%28pi%29%5E2 the volume is:
.
V+=+36%2A%28pi%29%5E2%2Asqrt%28pi%29
.
Hope this helps you to understand the relationship between the surface area and volume
equations for a given sphere.