Problem
Mortgage Rates The average 30-year fixed mortgage rate in the United States in the first week of May in 2010 through 2012 is approximated by \[ M(t)=\frac{55.9}{t^{2}-0.31 t+11.2} \] percent per year. Here $t$ is measured in years, with $t=0$ corresponding to the first week of May in 2010.+ (a) What was the average 30-year fixed mortgage rate in the first week of May in 2011 ( $t=1$ )? (Round your answer to two decimal places.) 4.7 () $\%$ (b) How fast was the 30-year fixed mortgage rate decreasing in the first week of May in 2011 ( $t=1)$ ? (Round your answer to two decimal places.) 5600 $\%$ per year
Solution
Step 1 ：The problem provides the function \(M(t)=\frac{55.9}{t^{2}-0.31 t+11.2}\) percent per year, where \(t\) is measured in years, with \(t=0\) corresponding to the first week of May in 2010.
Step 2 ：We are asked to find the average 30-year fixed mortgage rate in the first week of May in 2011, which corresponds to \(t=1\) in the given function.
Step 3 ：To find this, we substitute \(t=1\) into the function and calculate the result: \(M(1) = 4.701429772918419\).
Step 4 ：Rounding this to two decimal places, we get \(4.70\).
Step 5 ：Thus, the average 30-year fixed mortgage rate in the first week of May in 2011 was approximately \(\boxed{4.70\%}\).
