A castle built for the man who founded the town of Redstone, CO$,$ can now be yours, for $$19\mathrm{.}75$ million. $($Sherman$,$ T$.)$

A coal magnate, John Cleveland Osgood, built the castle for his wife near the town of Redstone, which was founded as a company town for people working in the coal mining industry. Construction began in $1897$ and was completed in $1902.($Sherman$,$ T$.)$

In the early $1900$s$,$ John Cleveland Osgood was known as the King of Coal in the West. He was sixth richest of the Robber Barons, a group of elite industrialists. As a social and industrial experiment, Osgood decided he wanted to build a town in the Colorado mountains that would provide better living conditions for the miners working nearby. Instead of the tents and tiny cabins common in mining towns, Osgood built Swiss$-$style cottages $$ with electricity and running water $-$ along the river. He also built a clubhouse with a theater, school, library, lodge, community garden and stables. $($Kesting$,$ A$.).$

Known as Redstone Castle and as the "Ruby of the Rockies," the $42-$room castle has almost $30,000$ square feet of interior space. It also has over $150$ acres of land, surrounded by beautiful scenery along the Crystal River. $($Sherman$,$ T$.)$

It's not the first time the castle has been up for sale in recent years. Steve and April Carver purchased it at an auction for $$2\mathrm{.}2$ million in $2016.($Oravetz$,$ J$.)$

The Carvers worked to renovate and restore the castle and have operated it as a hotel since $2018.($Oravetz$,$ J$.)$

It features nine restored boutique suites, where dignitaries such as Teddy Roosevelt and the Rockefellers once slept. It was also open to the public for guided tours on weekends and was available to host weddings and other private events. $($Oravetz$,$ J$.)$

Using PV and FV values in calculations:

$1.$ Calculate the annual compound growth rate of the Redstone Castle price since the house was purchased by Steve and April Carver $($in $2016)$ until it was listed for sale in $2020.($The growth rate should be calculated to two decimals in percentage form. Round the number of years to the whole number$).$

$2.$ Assume that the average compound growth rate prevailed since John Cleveland Osgood built the Castle in $1902$ until $2016$ was $3\mathrm{.}78\%$ per year compounded annually. Calculate the price of the house in $1902.($Round the number of years to the whole number$).($TIP: To get the answer correctly you need to use the price of the house in your calculations in dollars with all zeros$).$

$3.$ You were using the time value of money concept to answer question #$2.$ Think about the timeline for that problem. What year is considered as the time point $0$ in that problem?

$4.$ Assume that the average compound growth rate you were using in the question #$2$ now is compounded monthly. Calculate the price of the house in $1902.$ Compare with your answer to the question #$2.($Round the number of years to the whole number$).($TIP: To get the answer correctly you need to use the price of the house in your calculations in dollars with all zeros$).$

$5.$ Assume that the house was purchased for the price listed in $2020.$ A commercial bank loaned the buyer the whole price $($zero down payment$)$ for $30$ years. The loan must be repaid in equal monthly payments at the end of the month. The annual interest rate on the loan is $4\mathrm{.}37\%$ of the unpaid balance. What is the amount of the monthly payments?