Question

The average monthly rent for a? 1000-sq-ft apartment in a major metropolitan area from 1998 through 2005 can be approximated by the function below where t is the time in years since the beginning of 1998. Find the value of t when rents were increasing most rapidly. Approximately when did this? occur?f(t)= 1.6714t^4 - 22.45t^3 + 62.27t^2 + 6.108t + 1029The rents were increasing most rapidly when t = ?

Answer 1

t = 1.107

Explanation:

Finding the solution using derivatives involves finding the lower zero of the quadratic that is the second derivative of the given function. That second derivative will be ...

f''(t) = 12(1.6714)t^2 -6(22.45)t +2(62.27)

= 20.0568t^2 -134.7t +124.54

= 20.0568(t -3.35796)² -101.619 . . . . rewrite to vertex form

Then f''(t) = 0 when ...

t ≈ 3.35796 -√(101.619/20.0568) ≈ 1.10706

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The solution is perhaps more easily found using a graphing calculator to find the peak of the first derivative. (See attached.) It tells us ...

t ≈ 1.107

1.1 years after the beginning of 1998 is about 1.2 months into 1999.

Rents were increasing most rapidly in early February of 1999.