Question: The amount of money (in billions of dollars) lent to customers with credit scores below 620 for…

The amount of money (in billions of dollars) lent to customers with credit scores below 620 for subprime mortgages can be approximated by the function \( g(x) = 273.9e^{-0.19x} \), where \( x = 1 \) corresponds to the year 2001.

(a) Find the value of subprime mortgage lending in 2011 for the described customer base.

Solution

To find the value of subprime mortgage lending in 2011, we use the function \( g(x) = 273.9e^{-0.19x} \). Identify the year 2011. Since \( x = 1 \) corresponds to the year 2001, for 2011, \( x = 11 \). Substitute \( x = 11 \) into the function. \[ g(11) = 273.9e^{-0.19 \times 11} \] Calculate the exponent: \[ -0.19 \times 11 = -2.09 \] Substitute back: \[ g(11) = 273.9e^{-2.09} \] Now evaluate \( e^{-2.09} \): Using a calculator, \( e^{-2.09} \approx 0.124 \). Substitute back to find \( g(11) \): \[ g(11) = 273.9 \times 0.124 \] Finally, calculate: \[ g(11) \approx 33.9476 \] The value of subprime mortgage lending in 2011 is approximately 33.95 billion dollars.