Suppose a treacherous stretch of road winds through Deadman's Canyon for 5 miles.A highway department study estimates that better lighting would reduce the traffic fatality rates as follows: $$\begin{array} { c c c }
\text { Number of lights } & \text { Estimated deaths per year } & \text { Total installation cost (in millions) } \
0 & 3 & \$ 0 \mathrm {~m} \
1 & 2 & \$ 3 \mathrm {~m} \
2 & 1.5 & \$ 7 \mathrm {~m} \
3 & 1.2 & \$ 12 \mathrm {~m} \
4 & 1 & \$ 18 \mathrm {~m} \
5 & 0.9 & \$ 25 \mathrm {~m}
\end{array}$$
Suppose that experts have estimated the statistical value of a life saved to be $8m.The highway department should install ___ lights in Deadman's Canyon.

Question

Suppose a treacherous stretch of road winds through Deadman's Canyon for 5 miles.A highway department study estimates that better lighting would reduce the traffic fatality rates as follows: $$\begin{array} { c c c } 
\text { Number of lights } & \text { Estimated deaths per year } & \text { Total installation cost (in millions) } \
0 & 3 & \$ 0 \mathrm {~m} \
1 & 2 & \$ 3 \mathrm {~m} \
2 & 1.5 & \$ 7 \mathrm {~m} \
3 & 1.2 & \$ 12 \mathrm {~m} \
4 & 1 & \$ 18 \mathrm {~m} \
5 & 0.9 & \$ 25 \mathrm {~m}
\end{array}$$
Suppose that experts have estimated the statistical value of a life saved to be $8m.The highway department should install ___ lights in Deadman's Canyon.

Quizplus · Accepted Answer

To find the optimal number of lights, we need to minimize the total cost, which includes the cost of fatalities and installation. The cost of fatalities can be found by multiplying the estimated deaths per year by the value of a saved life, which is given to be $8m. The total cost is then the sum of the cost of fatalities and installation cost.

$$\begin{array} { c c c } 
\text { Number of lights } & \text { Cost of fatalities (in millions) } & \text { Total cost (in millions) } \
0 & 24 & 24 \
1 & 16 & 19 \
2 & 12 & 19 \
3 & 9.6 & 21.6 \
4 & 8 & 26 \
5 & 7.2 & 32.2
\end{array}$$

From the table, we can see that the total cost is minimized when two lights are installed, which costs $19m in total. Therefore, the best choice is option C, which suggests installing two lights.

Suppose a Treacherous Stretch of Road Winds Through Deadman's Canyon