With the current housing market situation in the United States, it is estimated that the number of houses built, \( H(t) \) (in millions) over the next 5 years, is related to the mortgage rate, \( r(t) \) (percent per year), and can be modeled by the equation: \[ 7 H(t)^{2}+r(t)=15 \] Find the rate of change of the number of houses being built with respect to time when the mortgage rate is \( 3 \% \) per year and is increasing at \( 0.1 \% \) per year. (Round your answer to the nearest whole number.) The number of houses built is?

To solve for the number of houses being built, \( H(t) \), when the mortgage rate \( r(t) \) is \( 3\% \) (i.e., \( r(t) = 3 \)), we start with the given equation: \[ 7 H(t)^{2} + r(t) = 15 \] Substituting \( r(t) = 3 \% \): \[ 7 H(t)^{2} + 3 = 15 \] Now, simplify this equation: \[ 7 H(t)^{2} = 15 - 3 \] \[ 7 H(t)^{2} = 12 \] \[ H(t)^{2} = \frac{12}{7} \] \[ H(t) = \sqrt{\frac{12}{7}} \] \[ H(t) \approx \sqrt{1.7143} \approx 1.309 \text{ million} \] Next, we find \( \frac{dH}{dt} \) when \( r(t) = 3\% \) and is increasing at \( 0.1\% \) per year, so \( \frac{dr}{dt} = 0.1 \). We take the derivative of both sides of the original equation with respect to time \( t \): \[ \frac{d}{dt}(7 H(t)^{2}) + \frac{dr}{dt} = 0 \] Using the chain rule on the first term: \[ 7 \cdot 2H(t) \frac{dH}{dt} + \frac{dr}{dt} = 0 \] Substituting \( H(t) \) and \( \frac{dr}{dt} \): \[ 14 H(t) \frac{dH}{dt} + 0.1 = 0 \] Solving for \( \frac{dH}{dt} \): \[ 14 (1.309) \frac{dH}{dt} + 0.1 = 0 \] \[ 14(1.309) \frac{dH}{dt} = -0.1 \] \[ \frac{dH}{dt} = \frac{-0.1}{14(1.309)} \approx \frac{-0.1}{18.326} \approx -0.00546 \text{ million/year} \] Now, rounding to the nearest whole number, the rate of change of the number of houses being built is approximately: \[ H(t) \approx 1 \text{ million houses built (rounded)} \] So, when rounding to the nearest whole number, the number of houses built is approximately \( 1 \) million.