What is the detailed solution to this problem?

Question

I've been trying to solve this problem from my Calculus textbook. ( the one attached as a pic ) the solutions page says that the answer is 550 Ib-ft

I couldn't solve it unfortunately. Please provide a step by step solution to this problem explaining which quantities were used in which integrations and why.

Thanks in advance

If you're wondering, the textbook is Calculus with Analytic Geometry by George Simmions 2ed Section 7.7 problem 5 page 249

Answer 1

$work = \int f\ dx$

$f(0) = 65\\ f(10) = 45\\ f(x) = 65-2x$

$\int_0^{10} 65-2x \ dx$

Answer 2

$$W=Fd$$ Total work = Work to lift bucket + Work to lift sand $\rightarrow W= Wb +Ws =Fb(d) + Fs(d)$

1.$Wb$ $$Fb=5lb$$ $$dWb=5dx$$ $$Wb= \int_0^{10} 5dx = 50 Lbft$$

2.$Ws$

Note that the sand weighs $60 lb$ at the beginning and $40 lb$ after being lifted $10m$, with the loss being uniform. Thus the force is unfiromly variable with respect to the position of the bucket, which means we need to formulate a linear equation for the force:

$$Fs=60-2x$$ You could check this equation by plugging in values for $x=0,5,10$ $$dWs=(60-2x)dx$$ $$Ws= \int_0^{10} (60-2x)dx = 500Lbft $$

$$W=500Lbft + 50Lbft = 550Lbft$$