$7.(20$ points$)$ You are asked to evaluate the possibility of using a larger diameter pipe in a line running from the source to the treatment plant. While the larger pipe will initially cost $\backslash (\backslash$$ $250,000\backslash )$ more than the originally planned pipe, it will reduce annual pumping costs. The pessimistic, most likely, and optimistic projections are $\backslash (\backslash$$ $9,000\backslash ),\backslash (\backslash$$ $17,000\backslash ),$ and $\backslash (\backslash$$ $22,000\backslash )$ respectively. The interest rate is $\backslash (6\backslash \%\backslash )$ and the design life of the water line is $40$ years. There is no salvage value.

a$.(10$ pts$)$ What is the present worth for each scenario?

b$.(5$ pts$)$ Use the range of estimates to compute the mean annual savings and then determine the present worth.

c$.(5$ pts$)$ Does the answer to part b match the most likely value from part a$?$ Why or why not?