Question

A study prepared for the National Association of Realtors estimates that in the southwest the number of housing starts N, in units of a million, over the next 5 years is related to the mortgage rate r, in percent per year, by the equation below.

9 N²+ r = 36
What is the rate of change of the number of housing starts with respect to time when the mortgage rate is 11% per year and its increasing at the rate of 1.5% per year?

Answer 1

The rate of change of the number of housing starts with respect to time when the mortgage rate is 11% per year and increasing at a rate of 1.5% per year is approximately -0.0008 million units per year.

Step-by-step explanation:

The given equation relating the number of housing starts N to the mortgage rate r is 9N² + r = 36. We want to find the rate of change of the number of housing starts with respect to time (dN/dt) when the mortgage rate is 11% per year and increasing at a rate of 1.5% per year.

To find the rate of change, we need to differentiate both sides of the equation with respect to time:

18NdN/dt + dr/dt = 0

Since we want to find dN/dt, we isolate dN/dt on one side:

18NdN/dt = -dr/dt

dN/dt = -dr/dt / (18N)

Substituting the given values, we have:

rate of change of the number of housing starts = -0.015 / (18 * 1.1) = -0.0008 million units per year