# How Many Laborers Should You Hire to Move Excess Topsoil from Your Farm?

• ploppers
In summary, the person needs to move excess topsoil from his farm and has the option to hire a truck and driver for $60/h. With a truck delivery time of 30 minutes and one person taking 40 hours to load the truck, the calculation for time is y = (40/x) + 0.5. To determine the number of workers needed, the time function is multiplied by the salary of the laborers and the driver, which results in the equation ((40/x) + 0.5)(18x + 30). However, there was a mistake made in the calculation and the correct solution is to hire 16 men, which would result in a total cost of$294.
ploppers

## Homework Statement

A person wants to move excess topsoil from his farm. He can hires a truck and driver for $60/h. The driver will take 30 mins to deliver a load of top soil. One person will take 40 h to load the truck with soil. Workers get$18/h (Including the time the truck takes). How many labourers should the person Hire?

Dunno :p

## The Attempt at a Solution

I figured that since it takes one person 40 hours, it should take two people 20 hours and 3 people 40/3 hours and so on...basically 40/x. That will represent the time it takes to move all the soil for one run. Then you must add the time for the driver to deliver with is 0.5 h. So the function for time is y = (40/x) + 0.5. That function will then be multiplied by the salary of both labourers and the driver which are 18x and 60. So i get:

= ((40/x) + 0.5)(18x) + ((40/x) + 0.5)(60)
= ((40/x) + 0.5)(18x + 30)
= 9x + 1200x^-1 + 735

Then I took the derivative and got
f'(x) = 9 -1200x^-2

To find the max slope should be 0

0 = 9 - 1200x^-2
1200/x^2 = 9
1200/9 = x^2

x = 11.54700

That does not match the answer in the back where it is 5 men that would make a total cost of $294 ploppers said: = ((40/x) + 0.5)(18x) + ((40/x) + 0.5)(60) = ((40/x) + 0.5)(18x + 30) = 9x + 1200x^-1 + 735 Isnt that s'posed to be $\ 9x+2400x^{-1}+c$ Solving i got x=16...doesnt help :( Anyone have a solution? Or did I make another mistake? As f(x) noted, you made a mistake here: ploppers said: = ((40/x) + 0.5)(18x) + ((40/x) + 0.5)(60) = ((40/x) + 0.5)(18x + 30) If you have stated the problem correctly, your book has made a mistake also. It will take 8 hours for 5 workers to load the truck, making the cost for the truck and driver alone$510. The workers would have to pay you $216 for the privilege of loading the truck to make the total cost$294.

I agree with f(x): Hire 16 men.

## 1. What is a rate of change?

A rate of change is a mathematical concept that refers to the measure of how one variable changes in relation to another variable. It is often represented as a ratio of the change in the dependent variable to the change in the independent variable.

## 2. How do you calculate a rate of change?

To calculate a rate of change, you need to determine the change in the dependent variable and the change in the independent variable over a given time period. Then, divide the change in the dependent variable by the change in the independent variable to get the rate of change.

## 3. What is the difference between average rate of change and instantaneous rate of change?

The average rate of change is calculated over a specific interval of time, while the instantaneous rate of change is calculated at a specific point in time. The average rate of change provides an overall view of how a variable changes over time, while the instantaneous rate of change gives the exact rate of change at a specific moment.

## 4. How is rate of change used in science?

In science, rate of change is used to analyze and understand how different variables are related to each other. It is particularly useful in studying the behavior of natural phenomena, such as changes in temperature, growth rate of organisms, or rate of chemical reactions.

## 5. What are some real-life examples of rates of change?

There are many real-life examples of rates of change, such as the speed of a car on a highway, the rate of population growth in a city, the change in stock prices over time, and the rate of evaporation in a pond. Any situation where there is a change in a variable over time can be represented and analyzed using rates of change.