The number of banks in a country for the years 1935 through 2009 is given by the following function. \( f(x)=\left\{\begin{array}{r}84.8 x+12,365 \text { if } x<90 \\ -376.3 x+48,686 \text { if } x \geq 90\end{array}\right. \), where \( x \) is the number of years after 1900 Complete parts (a)-(b). a) What does this model give as the number of banks in \( 1970 ? 1990 ? 2020 \) ? The number of banks in 1970 is

We begin by noting that in our function \( f(x) \), the variable \( x \) represents the number of years after 1900. Thus, to find the number of banks in any given year, we first convert the year to the appropriate \( x \) value. **For 1970:** - Compute \( x \) for 1970 as: \[ x = 1970 - 1900 = 70 \] - Since \( 70 < 90 \), we use the first part of the piecewise function: \[ f(x) = 84.8x + 12,\!365 \] - Plug in \( x = 70 \): \[ f(70) = 84.8 \times 70 + 12,\!365 \] - Calculate \( 84.8 \times 70 \): \[ 84.8 \times 70 = 5,\!936 \] - Then add: \[ f(70) = 5,\!936 + 12,\!365 = 18,\!301 \] Thus, the number of banks in 1970 is \( 18,\!301 \). **For 1990:** - Compute \( x \) for 1990: \[ x = 1990 - 1900 = 90 \] - Since \( 90 \geq 90 \), we use the second part of the function: \[ f(x) = -376.3x + 48,\!686 \] - Plug in \( x = 90 \): \[ f(90) = -376.3 \times 90 + 48,\!686 \] - Calculate \( 376.3 \times 90 \): \[ 376.3 \times 90 = 33,\!867 \] - Then subtract: \[ f(90) = 48,\!686 - 33,\!867 = 14,\!819 \] Thus, the number of banks in 1990 is \( 14,\!819 \). **For 2020:** - Compute \( x \) for 2020: \[ x = 2020 - 1900 = 120 \] - Since \( 120 \geq 90 \), again we use the second part of the function: \[ f(x) = -376.3x + 48,\!686 \] - Plug in \( x = 120 \): \[ f(120) = -376.3 \times 120 + 48,\!686 \] - Calculate \( 376.3 \times 120 \): \[ 376.3 \times 120 = 45,\!156 \] - Then subtract: \[ f(120) = 48,\!686 - 45,\!156 = 3,\!530 \] Thus, the number of banks in 2020 is \( 3,\!530 \). **Summary:** - 1970: \( 18,\!301 \) banks - 1990: \( 14,\!819 \) banks - 2020: \( 3,\!530 \) banks Since the question asks, "The number of banks in 1970 is", the answer is: \[ 18,\!301 \]