Question

Ken can hire a dump truck for $40/h (driver included). The truck takes 30 minutes to deliver a load of topsoil and return. It would take 40 hours for one person to load the truck with topsoil. Labourers get $10/h whether they are loading the truck or standing idle while they await its return. How many labourers should be hired to minimize the cost per load? What is the minimum cost?

Answer 1

i get 4sqrt2 as number of laborers to minimize cost per load...but not 100%
lets see if @tiaph agrees with me

Answer 2

The answer given in the back of the book is 18 labourers... i just have no idea how to get there...

Answer 3

Total cost to load one truck worth of topsoil, unload and return, C= ((40/x)*10)+ (40 * (40/x)) + 40*0.5 + 10x*0.5
where
x is the number of workers.
((40/x)*10) is how much the workers are paid in total when digging
(40*(40/x)) is how much the driver is paid when workers are digging
40*0.5 is how much the driver is paid when delivering the load
10x *0.5 is how much the workers is paid when the driver is delivering.

Simplying the expression,
C= 400/x + 1600/x + 5x + 20
C= 2000/x + 5x +20
dC/dx = -2000/x^2 + 5
Stationary point, equate dC/dx = 0,
-2000/x^2 +5 = 0
-2000/x^2 = -5
x= 20

lol i got 20 labourers. I cant figure out where im wrong though

Answer 4

oh wait i think i did it right, just made a stupid computational error
i get 17.88 which they round up to 18
Let n be num of workers, T is time to complete 1 load
C(t) = 40t +10nt
T = (40/n) + 1/2 --> plug this is for time

\[C(n) = 40(\frac{40}{n}+\frac{1}{2}) +10n(\frac{40}{n}+\frac{1}{2})\]
\[C(n) = 420+5n+\frac{1600}{n}\]
differentiate and set equal to 0 to minimize C(n)
\[\frac{dC}{dn} = 5-\frac{1600}{n^{2}} = 0\]
\[\rightarrow 5n^{2} = 1600\]
\[\rightarrow n = \sqrt{320} = 17.88\]

Answer 5

@tiaph , you forgot the x when looking at workers digging cost
--> (40/x)*10 should be (40/x)*10x

Answer 6

YEAAAA thanks, i have been staring at it, wondering where i gone wrong. thanks for saving me so much pain.

Answer 7

Whoa! Thanks for this solution! I'm gonna try and work it out now and see if i get the same.

Answer 8

yep, your welcome