Question

A study for the National Association of Realtors estimates the number of housing starts per year as "N(r) = 7 / (1 + 0.02r^2)" million units, where "r" (percent) is the mortgage rate. If the mortgage rate "r(t) = (5t + 75) / (t + 10)" percent/year, find an expression for the number of housing starts per year as a function of "t," "t" months from now.

Answer 1

To find the number of housing starts as a function of time, substitute the expression for the mortgage rate r(t) over time into the given function N(r). The resulting function N(t) expresses housing starts in terms of months from now.

Step-by-step explanation:

The task is to find an expression for the number of housing starts per year as a function of time t, months from now, incorporating the given mortgage rate as a function of time. We have been given:

N(r) = \( \frac{7}{1 + 0.02r^2} \) million units, where r is the mortgage rate.

The mortgage rate, r(t), varies with time according to the formula:

r(t) = \( \frac{5t + 75}{t + 10} \)% per year.

Substituting the expression for r(t) into N(r), we obtain:

N(t) = \( \frac{7}{1 + 0.02\left(\frac{5t + 75}{t + 10}\right)^2} \) million units.

This function describes the number of housing starts based on time t and changing mortgage rates.