Piston Loading

Consider a steam engine's piston has a 10 in diameter, and a 1 in diameter piston rod. If the piston's rod has a design stress of 1,000 psi, what is the max steam gauge pressure in psi?

Expand Hint
$$$Pressure =\frac{Force}{Area}$$$
Hint 2
Since the piston force is the same force on the piston's rod, uniaxial loading is applied:
$$$\sigma =\frac{F}{A}$$$
where $$\sigma$$ is the stress on the cross section, $$F$$ is the loading, and $$A$$ is the cross sectional area.
Uniaxial loading:
$$$\sigma =\frac{F}{A}$$$
where $$\sigma$$ is the stress on the cross section, $$F$$ is the loading, and $$A$$ is the cross sectional area.
$$$F_{rod}=\sigma A_{rod}=(1,000\frac{lb}{in^2})(\frac{\pi}{4})(1in)^2=785\:lb$$$
To find the force on the piston caused by the steam pressure:
$$$\sum F=(pressure)(area)=(P)(\frac{\pi}{4} \cdot D^2)$$$
$$$F_{piston}=(P_{gauge})[\frac{\pi}{4} \cdot (10in)^2]$$$
Because $$F_{piston}=F_{rod}$$ :
$$$785lb=(P_{gauge})[\frac{\pi}{4} \cdot 100in^2]$$$
Finding the max steam gauge pressure:
$$$P_{gauge}=\frac{785lb}{\frac{\pi}{4} \cdot 100in^2}=10\:psi$$$
10 psi