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Please explain in the simplest way you know how.

Q: A water tank in the shape of a cylinder of radius 6ft and length 15ft is lying on its side. It's half full of water, weighing 62.5lb/ft^3. How much work is required to pump all the water if the pump is 9ft above the top edge of the tank?

Solution: Drawing the picture and placing the sideways tank directly on the x-axis.

We want to find a generic cross-sectional slice. So we need the volume of a rectangle.

V = LWH

V = (15)(W)(dy)

And so now we need to isolate W. To do this we notice that a slice of the cylinder is a rectangle on the x-axis, and half of the width is one side of the x-axis, and the other half is the other side of the x-axis. So 2x is the width of the slice of the cylinder. But now we need to determine what 'x' is? This is where I get lost.

to get one x by itself we use the equation for a circle: x^2 + y^2 + r^2

But how does it become this afterwards:

x^2 + (y - 6)^2 = 36?

I don't understand where the (y-6) came from?

I understand the rest of the problem and can solve. It's just this one aspect I don't get at all.

Thank you

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The equation $x^2 + (y - 6)^2 = 36$ is the equation for a circle with radius $6$ centered at the point $(0,6)$. Using this equation assumes that the ground is level with the $x$ axis, so the tank's center is actually $6$ ft above the ground and the $x$ axis.

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