Answer
a) $200$ lb/in
b) $400$ lb-in $\approx 33.3$ ft-lb
c) 8 in
Work Step by Step
a) Since, we know $F=k x$
Then, $k=\dfrac{F}{x}=\dfrac{800}{4}= 200$ lb/in
b) Now, integrate.
From part (a) $k =200$
$\int_0^2 kx dx= 200[ \dfrac{x^2}{2}]_0^2$
or, $100[ 2^2-0] =400 lb- in \approx 33.3 ft -lb$
c) Since, we know $F=k x$ and here, F=1600 N
Then, $1600=(200) (x) \implies x=\dfrac{1600}{200}$
$x=8$ inches