A Simple Calculus Question I Can't Get

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In summary, Stan needs to move excess topsoil and is trying to minimize the cost per load by hiring labourers to load the truck and a driver for delivery. The equation for the total cost is C = 60h + (18x)h, where h represents the number of hours for one complete load and x represents the number of labourers hired. After solving for the minimum cost, the answer is approximately 16 labourers. However, the back of the book claims the answer is 5, which is most likely due to an error in the problem statement.
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Sane
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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.

[tex]C = 60h + (18x)h[/tex]

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.

[tex]h = \frac{40}{x} + \frac{1}{2}[/tex]

Therefore, the solution to the equation is:

[tex]\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*}[/tex]

Plugging x back into the original equation will prove a minimum cost. However, the back of the book says the answer is 5. Why?
 
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  • #2
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
 
  • #3
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!
 

Related to A Simple Calculus Question I Can't Get

1. What is calculus?

Calculus is a branch of mathematics that studies continuous change and the properties of functions. It includes the concepts of differentiation and integration, which are used to analyze rates of change and find solutions to problems involving curves and surfaces.

2. What is the difference between differential and integral calculus?

Differential calculus deals with the rate of change of a function, while integral calculus involves finding the area under a curve. In other words, differential calculus focuses on finding the derivative of a function, while integral calculus deals with finding the antiderivative.

3. Why is calculus important?

Calculus is a fundamental tool in many fields, including physics, engineering, economics, and computer science. It allows us to model and understand complex systems and make accurate predictions about their behavior. It is also essential for solving optimization problems and finding precise solutions to real-world problems.

4. How do I solve a simple calculus problem?

To solve a calculus problem, you need to identify the given function and the type of problem (e.g., finding the derivative or integral). Then, use the appropriate formulas and techniques to find the solution. It is important to have a strong understanding of the fundamental concepts of calculus, such as limits, derivatives, and integrals, to solve problems effectively.

5. What are some common mistakes made in calculus?

Some common mistakes in calculus include forgetting to use the chain rule when differentiating composite functions, not understanding the concept of limits, and making algebraic errors. It is crucial to pay attention to detail and practice regularly to avoid these mistakes and improve your skills in calculus.

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