### Video Transcript

The ratio between the cross section
of the large piston to that of the small piston of a hydraulic lift is three. By how much should the pressure
acting on the large piston increase to maintain the two pistons at the same
horizontal level if the pressure acting on the small piston increased by ฮ๐? (A) Three ฮ๐, (B) ฮ๐, (C) ฮ๐
over three, (D) two ฮ๐.

Here, we are told that there is a
hydraulic lift and the ratio between the area of the large piston to the small
piston is three. We are then asked, if the pressure
acting on the small piston increases by ฮ๐, by how much should the pressure acting
on the large piston increase so that the two pistons stay at the same horizontal
level?

Letโs remind ourselves about
Pascalโs principle, information about the pressure fluids exert, and some general
properties of hydraulics. Pascalโs principle states that at
any point in a fluid, the pressure exerted by the fluid at that point is equal in
all directions. As well, remember that the pressure
exerted by a fluid at a point is due to the weight of the water directly above that
point. This means that if we have two
points that are at equal depths, they will be at the same pressure as long as the
only force on them is the weight of the fluid above.

A hydraulic lift, like we see in
this diagram, is constructed to utilize these properties of fluids. When a force is applied to the
small piston, a force is transferred through the fluid and pushes up on the larger
piston. We can find the pressure applying
to each of the pistons using this equation. The pressure ๐ exerted over an
area ๐ด is equal to the force applied ๐น divided by that area. This means that the pressure on the
base of the small piston is equal to the force being applied divided by the area of
the small piston. The same equation is used for the
large piston as well. The pressure on the base of the
large piston is equal to the force of the fluid pushing up divided by the area of
the large piston.

Now, because the fluid contained
between the two pistons is completely enclosed and the same, we can set the pressure
on the base of these two pistons to be the same as well. This allows us to relate the
magnitudes of these forces to the cross-sectional areas of the pistons, like so. The force acting on the small
piston divided by the area of its face is equal to the force acting on the large
piston divided by its area. Notice that the left-hand side of
this equation is equal to the pressure over the area of the large piston and the
right-hand side is equal to the pressure over the area of the small piston.

Now, since the pressures must be
equal to each other, if the pressure on the small piston increases by an amount ฮ๐,
in order for the pistons to stay at the same horizontal level, and for this equation
to stay balanced, the pressure on the larger piston must also increase by the same
amount. Therefore, the larger piston will
also need an increase in pressure of ฮ๐. The difference in area between the
two pistons doesnโt matter here, since theyโre at the same horizontal level. So, the second option, ฮ๐, is the
correct answer.