Questions: The concrete plant is located 12 miles from one end of the job site. The trucks carrying the concrete to the site can travel at 35 mph to the site and 45 mph on the return trip and can carry 20 tons of material. How many cy of material is 20 tons? It takes 1.5 minutes to load each truck and 2.25 minutes to unload the material into the paver. How many trucks do you need to keep up with the operation? How much will the haul operation cost?

Solution Steps

To convert 20 tons of material to cubic yards, we use the density of the material. Given that the density is \( 1.5 \, \text{tons/cubic yard} \), the conversion is calculated as follows:

\[ \text{cubic yards} = \frac{\text{tons}}{\text{density}} = \frac{20 \, \text{tons}}{1.5 \, \text{tons/cubic yard}} = \frac{20}{1.5} = \frac{40}{3} \, \text{cubic yards} \]

The round trip time for a truck includes the time to travel to the site, the time to return, and the loading and unloading times. The travel times are calculated as follows:

\[ \text{travel time to site} = \frac{\text{distance}}{\text{speed to site}} = \frac{12 \, \text{miles}}{35 \, \text{mph}} = \frac{12}{35} \, \text{hours} \]

\[ \text{travel time return} = \frac{\text{distance}}{\text{speed return}} = \frac{12 \, \text{miles}}{45 \, \text{mph}} = \frac{12}{45} \, \text{hours} \]

The total round trip time is then:

\[ \text{round trip time} = \text{travel time to site} + \text{travel time return} + \text{loading time} + \text{unloading time} \]

Substituting the values:

\[ \text{round trip time} = \frac{12}{35} + \frac{12}{45} + \frac{1.5}{60} + \frac{2.25}{60} \]

To find the number of trucks required, we first calculate how many trips a single truck can make during an 8-hour operation:

\[ \text{trips per truck} = \frac{\text{operation time}}{\text{round trip time}} = \frac{8 \, \text{hours}}{\text{round trip time}} \]

Next, we determine the total number of trips needed based on the cubic yards required:

\[ \text{total trips needed} = \frac{\text{cubic yards}}{\text{tons per truck}} = \frac{\frac{40}{3}}{20} = \frac{2}{3} \]

Finally, the number of trucks needed is calculated as:

\[ \text{trucks needed} = \frac{\text{total trips needed}}{\text{trips per truck}} \]

The total distance traveled by all trucks is given by:

\[ \text{total distance} = 2 \times \text{distance} \times \text{total trips needed} = 2 \times 12 \, \text{miles} \times \frac{2}{3} = 16 \, \text{miles} \]

The haul operation cost is then calculated as:

\[ \text{haul cost} = \text{total distance} \times \text{cost per mile} = 16 \, \text{miles} \times 2.5 \, \text{dollars/mile} = 40 \, \text{dollars} \]

Final Answer