SOLUTION: The formula for the surface area of a sphere is {{{A=4*pi*r^2}}}, where A is surface area and r is the radius of the sphere a) use the formula {{{A=4*pi*r^2}}} to express

Algebra ->  Surface-area -> SOLUTION: The formula for the surface area of a sphere is {{{A=4*pi*r^2}}}, where A is surface area and r is the radius of the sphere a) use the formula {{{A=4*pi*r^2}}} to express       Log On


   

Question 83801: The formula for the surface area of a sphere is
A=4%2Api%2Ar%5E2, where A is surface area and r is the
radius of the sphere


a) use the formula A=4%2Api%2Ar%5E2 to express r in terms of A and
rationalize the denominator. Explain your answer


b) suppose the surface area of a child's ball is
about 113 square inches. Find the radius of the
ball and explain your steps.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The formula for the surface area of a sphere is
A=4%2Api%2Ar%5E2, where A is surface area and r is the
radius of the sphere


a) use the formula A=4%2Api%2Ar%5E2 to express r in terms of A and
rationalize the denominator. Explain your answer
Since A=4%2Api%2Ar%5E2 we can solve for r by these steps

A=4%2Api%2Ar%5E2 Start with the given equation

A%2F%284%2Api%29=cross%28%284%2Api%29%2F%284%2Api%29%29r%5E2 Divide both sides by 4%2Api

sqrt%28A%2F%284%2Api%29%29=r Take the square root of both sides

sqrt%28A%29%2Fsqrt%28%284%2Api%29%29=r Break up the square root using the identity sqrt%28x%2Fy%29=sqrt%28x%29%2Fsqrt%28y%29}

sqrt%28A%29%2F%28sqrt%284%29%2Asqrt%28pi%29%29=r Break up the square root using the identity sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29}

sqrt%28A%29%2F%282%2Asqrt%28pi%29%29=r Take the square root of 4 to get 2


%28sqrt%28pi%29%2Fsqrt%28pi%29%29%28sqrt%28A%29%2F%282%2Asqrt%28pi%29%29%29=r Now rationalize the denominator by multiplying the numerator and denominator by sqrt%28pi%29. This is equivalent to %28sqrt%28pi%29%29%5E2=pi which in other words eliminates the square root in the denominator


%28sqrt%28pi%29sqrt%28A%29%2F%282%2Api%29%29=r Multiply and reduce

sqrt%28pi%2AA%29%2F%282%2Api%29=r Combine the square roots in the numerator

So now we have an equation in terms of A which is

r=sqrt%28pi%2AA%29%2F%282%2Api%29



b) suppose the surface area of a child's ball is
about 113 square inches. Find the radius of the
ball and explain your steps.


Since we have an equation in terms of A, we can plug in the given surface area (113 sq in) in for A and solve for r

r=sqrt%28pi%2A113%29%2F%282%2Api%29 Plug in A=113

Now lets approximate pi to 3.14. From here on, all of our values will be approximations. So even though I don't show it (I'm unable to), keep this in mind.

r=sqrt%283.14%2A113%29%2F%282%2A3.14%29 Replace pi with 3.14

r=sqrt%28354.82%29%2F%286.28%29 Multiply

r=18.837%2F%286.28%29 Take the square root

r=2.999 Divide

So the for a ball with a surface area of 113 sq in., the radius is about 3 inches