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Will the water pressures at the base of two different sized cylinders be the same if they are filled to the same height?

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Yes the pressure is the same, as the pressure is given by $pressure=(density)(depth)(gravitational$ $acceleration)=\rho h g$

This can be shown to be true for your cylinders, if we have the larger one to have radius $r_1$, and the smaller $r_2$, then the mass of the water, in each cylinder, is $m_1=\pi h \rho(r_1)^2 $ for the larger cylinder, and $m_2=\pi h \rho(r_2)^2 $ for the smaller cylinder.

Now the force pushing on the base of the cylinders, is equal to the gravitational force of the water. therefore the force on the base of cylinder the larger cylinder is $f_1=m_1g=gh \rho\pi (r_1)^2$ and the force for the smaller is $f_2=m_2g=gh \rho\pi (r_2)^2$.

Finally the pressure is given by $P=\frac{F}{A}$ where $A$ is the area that the force is pushing against. so the area for the larger cylinder is $P_1=\frac{F_1}{\pi (r_1)^2}=\frac{gh \rho\pi (r_1)^2}{\pi (r_1)^2}=gh \rho$ and for the smaller cylinder $P_2=\frac{F_2}{\pi (r_2)^2}=\frac{gh \rho\pi (r_2)^2}{\pi (r_2)^2}=gh \rho$ therefore $P_1=P_2$

So the pressure is the same for both cylinders.

hopefully that helps

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