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A design on the surface of a balloon is 5 cm wide when the balloon holds 71 cm^3 of air. How much air does the balloon hold when the design is 10 cm wide? Explain the method you use to find the amount of air.

 Apr 6, 2016
 #1
avatar+129852 
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If the design width doubles, then I'm assuming that the surface area of the balloon doubles when the balloon is inflated a second time.  And the balloon can be considered as a sphere.

 

Let's let the original radius = r

 

Then, the original surface area of the balloon =  4pi *r^2

 

Then, for the suface area to double, the radius must increase by a factor of  √2

 

Proof :   4pi (√2 r)^2  = 4pi (2)r^2  = 8pi*r^2  = twice the original surface area

 

So...let's find the original radius using the volume for a sphere

 

71  =(4/3)pi*r^3     divide both sides by (4/3)pi

 

71/ [ (4/3)pi] = r^3       take the cube root of both sides

 

cube root ( 71/ [ (4/3)pi] ) = r   = about 2.5687cm^3

 

So......if the original radius increases by a factor of √2, the new volume is :

 

V = (4/3)pi (2.5687√2)^3  = about 200.8 cm^3

 

 

 

cool cool cool

 Apr 6, 2016
 #2
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THANK YOU!

Carpe♥Diem  Apr 6, 2016

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