ANSWER
1250 N
EXPLANATION
We have a hydraulic press, which is something like this:
The pressure of the fluid inside the press is the same on both pistons. The pressure is related to the force and the area of the piston by,
[tex]P=\frac{F}{A}[/tex]As stated before, P1 = P2,
[tex]\frac{F_1}{A_1}=\frac{F_2}{A_2}[/tex]We know the diameter of the pistons d1 = 8cm and d2 = 40cm, and the force applied to the small piston is F1 = 50N. We have to solve the equation above for F2,
[tex]F_2=F_1\cdot\frac{A_2}{A_1}[/tex]The pistons are circular, thus the area is.
[tex]A=\pi\cdot r^2=\pi\cdot(\frac{d}{2})^2=\pi\cdot\frac{d^2}{4}_{}[/tex]Replace into the equation of the force,
[tex]F_2=F_1\cdot\frac{\pi\cdot\frac{d^2_2}{4}}{\pi\cdot\frac{d^2_1}{4}}[/tex]π/4 cancels out,
[tex]F_2=F_1\cdot\frac{d^2_2}{d^2_1}[/tex]Usually, we have to put all the units so that they are consistent - in this case, that would mean to convert the diameters to meters, but since the diameters are dividing their units will cancel out, so we don't need to convert those measures from centimeters to meters.
Replace F1, d1, and d2 for their values,
[tex]F_2=50N\cdot\frac{40^2cm^2}{8^2cm^2}[/tex][tex]F_2=50N\cdot25=1250N[/tex]The force in the bigger piston is 1250 N.
