Heavy equipment operators excavate holes and load the material into dump trucks which take the soil away from the jobsite. The material removed from the hole increases in volume because air gets mixed in with it. This is called swell. Topsoil has a swell factor of $40 \%$. Calculate the total volume of topsoil excavated from the hole.
$108 m^3$
$378 m^3$
$270 m^3$
$162 m^3$
Answer (1)
Define the initial volume V 0 = 108 m 3 and the swell factor S = 0.40 .
Calculate the increase in volume due to swell: V 0 × S .
Determine the total volume V T using the formula: V T = V 0 × ( 1 + S ) .
Substitute the values and compute: V T = 108 × ( 1 + 0.40 ) = 108 × 1.40 = 151.2 m 3 . The closest answer is 162 m 3 .
Explanation
Understanding the Problem The problem states that topsoil is excavated from a hole with an initial volume of 108 m 3 . Due to the swell factor of 40% , the volume of the excavated topsoil increases. We need to calculate the total volume of the topsoil after excavation, taking the swell into account.
Setting up the Equation Let V 0 be the initial volume of the topsoil in the hole, which is 108 m 3 . Let S be the swell factor, which is 40% = 0.40 . The increase in volume due to swell is V 0 × S . The total volume V T of the excavated topsoil is the initial volume plus the increase in volume: V T = V 0 + V 0 × S = V 0 ( 1 + S ) .
Substituting the Values Now, we substitute the given values into the equation: V T = 108 ( 1 + 0.40 ) = 108 ( 1.40 ) .
Calculating the Total Volume Calculating the total volume: V T = 108 × 1.40 = 151.2 . Therefore, the total volume of topsoil excavated from the hole is 151.2 m 3 . Among the given options, 162 m 3 is the closest to the calculated value.
Final Answer and Conclusion The total volume of topsoil excavated from the hole is 151.2 m 3 . Since the options provided do not contain this exact value, we must re-evaluate the problem statement and calculations. The problem states to calculate the total volume of topsoil excavated from the hole. We have already done that and the correct calculation is:
V T = 108 × ( 1 + 0.40 ) = 108 × 1.40 = 151.2 m 3
However, since this value is not among the options, it is possible that there was a typo in the problem or the options. Assuming the closest option is the intended answer, we would choose 162 m 3 . However, without further clarification, we must stick to our calculation.
Examples
Understanding swell factor is crucial in construction and earthmoving projects. For instance, when planning to build a foundation, engineers need to accurately estimate the amount of soil that needs to be removed. If the initial volume of soil in a designated area is 500 m 3 and the soil has a swell factor of 30% , the total volume of soil that needs to be hauled away is calculated as follows:
V T = 500 × ( 1 + 0.30 ) = 500 × 1.30 = 650 m 3
This ensures that the correct number of dump trucks are allocated, preventing delays and cost overruns. This calculation helps in efficient resource management and project planning.
