A newly constructed bridge costs $$10,000,000.$ The same bridge is estimated to need renovation every $10$ years at a cost of $$1,000,000.$ Annual repairs and maintenance are estimated to be $$100,000$ per year.

a$)$ If the interest rate is $5\%,$ determine the capitalized$-$equivalent cost of the bridge.

b$)$ Suppose that the bridge must be renovated every $15$ years, not every $10$ years. What is the capitalized cost of the bridge if the interest rate is the same as in $($a$)?$

c$)$ Repeat $($a$)$ and $($b$)$ with an interest rate of $10\%.$ What can you say about the effect of interest on the results?