Questions: Suppose that the amount of time it takes to build a highway varies directly with the length of the highway and inversely with the number of workers. Suppose also that it takes 100 workers 6 weeks to build 4 miles of highway. How many miles of highway could 160 workers build in 15 weeks? miles

Solution Steps

To solve this problem, we need to use the concept of direct and inverse variation. The time to build the highway varies directly with the length and inversely with the number of workers. We can set up a relationship using a constant of proportionality. First, find the constant using the given data, then use it to find the unknown length of the highway for the new conditions.

The time \( T \) to build a highway varies directly with the length \( L \) of the highway and inversely with the number of workers \( W \). This can be expressed as: \[ T = k \cdot \frac{L}{W} \] where \( k \) is a constant of proportionality.

Using the given data where \( W = 100 \), \( T = 6 \) weeks, and \( L = 4 \) miles, we can substitute these values into the equation to find \( k \): \[ 6 = k \cdot \frac{4}{100} \] Solving for \( k \): \[ k = 6 \cdot \frac{100}{4} = 150 \]

Now, we need to find out how many miles of highway \( L \) can be built with \( W = 160 \) workers in \( T = 15 \) weeks. Using the same relationship: \[ 15 = 150 \cdot \frac{L}{160} \] Rearranging to solve for \( L \): \[ L = \frac{15 \cdot 160}{150} = 16 \]

Final Answer