A Simple Calculus Question I Can't Get

[Sane]

I don't know why ... I've done every single question in the textbook up to this point, and am now stumped on one of the easier ones. I've made my equation, and solved it, but it is not the correct answer. Nor does the correct answer make any sense.

Either the back of the book is incorrect, or I have clearly misread the question. I'll show you what I did...

Stan needs to move some excess topsoil from his farm. He can hire a dump truck and a driver for $60/h. The driver will take 30 min to deliver a load of topsoil and return to the farm. One person will take 40 h to load the truck with soil. Labourers get $18/h (whether they are loading the truck with soil or waiting for the truck to return). How many labourers should Stan hire to minimize the cost per load?

Let h represent the number hours that one complete load will take. Let x represent the number of labourers that Stan is hiring to load the truck with soil. Let C represent the total cost of completing one load.

$$C = 60h + (18x)h$$

Since the number of hours is determined by the number of labourers, plus half an hour of delivering the soil, the equation for h is as follows.

$$h = \frac{40}{x} + \frac{1}{2}$$

Therefore, the solution to the equation is:

$$\begin{align*} C &= 60(\frac{40}{x} + \frac{1}{2}) + (18x)(\frac{40}{x} + \frac{1}{2})\\ &= \frac{2400}{x} + 30 + 720 + 9x\\ \frac{dC}{dx} &= \frac{-2400}{x^{2}} + 9\\ x^{2} &= \frac{2400}{9}\\ x &\approx 16\\ \end{align*}$$

Plugging x back into the original equation will prove a minimum cost. However, the back of the book says the answer is 5. Why?

 

[Gargan]

As you have stated the problem, your answer is correct. I don't see how the correct answer could be 5, unless the problem isn't stated clearly enough in the book. I have had a few textbooks with answers that are incorrect in the back, so I think this is what has happened

 

[Sane]

Yes, I believe you are correct. Another friend from my class says the back of the book must be wrong. I'm glad that was the problem. Thanks for looking into it!