A pool is shaped like a cylinder, and has a volume of 125(pi) cubic feet. What is the volume of another pool has the same height but doubled diameter?
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say the height is 5 and the diameter is 10, the the volume would be \(\pi\times r^2h=\pi \times 5^2\times 5=125\pi\)
since half the diameter is the radius
double that and you would have a diameter of 20 making the radius 10
compute \[\pi\times 10^2\times 5\]
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we can do it without the numbers i chose if you like
no i doubled the diameter from 10 to 20 so the radius got doubled from 5 to 10
lets do it another way, without the made up numbers ok?
when you compute the volume it is the area of the circle times the height
the area goes with the square, as in \(A=\pi r^2\)
therefore, if you double the diameter i.e. multiply it by 2 the area of the circle is multiplied by \(2^2=4\)
in other words the area will be 4 times as large making the volume also 4 times as large (since the height is fixed)
so do I multiply \[125\pi \times4\] ?
which is the exact same thing you would get if you computed \[10^2\times 5\times \pi\]