If \( f(x) \) is an exponential function of the form of \( y=a b^{x} \) where \( f(3.5)=21 \) and \( f(8.5)=37 \), then find the value of \( f(14) \), to the nearest hundredth.

To solve the problem, we first set up the equations based on the information given. The function is of the form \( f(x) = a b^x \). 1. From \( f(3.5) = 21 \): \[ ab^{3.5} = 21 \quad \text{(1)} \] 2. From \( f(8.5) = 37 \): \[ ab^{8.5} = 37 \quad \text{(2)} \] Next, we can divide equation (2) by equation (1) to eliminate \( a \): \[ \frac{ab^{8.5}}{ab^{3.5}} = \frac{37}{21} \] This simplifies to: \[ b^{8.5 - 3.5} = \frac{37}{21} \Rightarrow b^5 = \frac{37}{21} \] Now, let's find \( b \): \[ b = \left( \frac{37}{21} \right)^{\frac{1}{5}} \] Next, we substitute \( b \) back into equation (1) to find \( a \). We can rewrite equation (1) as: \[ a = \frac{21}{b^{3.5}} \] Substituting in our expression for \( b \): \[ a = \frac{21}{\left( \left(\frac{37}{21}\right)^{\frac{1}{5}} \right)^{3.5}} \] This becomes: \[ a = \frac{21}{\left( \frac{37}{21} \right)^{\frac{3.5}{5}}} = 21 \cdot \left( \frac{21}{37} \right)^{\frac{3.5}{5}} \] To find \( f(14) \): \[ f(14) = ab^{14} \] Using \( a \) and \( b \) we found earlier: \[ f(14) = \left( 21 \cdot \left( \frac{21}{37} \right)^{\frac{3.5}{5}} \right) \cdot \left( \frac{37}{21} \right)^{\frac{14}{5}} \] Calculating \( b^{14} \): \[ b^{14} = \left( \frac{37}{21} \right)^{\frac{14}{5}} \] Substituting this into our equation gives: \[ f(14) = 21 \cdot \left( \frac{21}{37} \right)^{\frac{3.5}{5}} \cdot \left( \frac{37}{21} \right)^{\frac{14}{5}} \] Now, we can find a simplified expression for \( f(14) \): \[ f(14) = 21 \cdot \frac{21^{\frac{14}{5}}}{37^{\frac{14}{5}}} \cdot \frac{37^{\frac{3.5}{5}}}{21^{\frac{3.5}{5}}} = 21^{1 + \frac{14}{5} - \frac{3.5}{5}} \cdot 37^{\frac{3.5}{5} - \frac{14}{5}} \] Calculating the new powers: \[ = 21^{(1 + \frac{14}{5} - \frac{3.5}{5})} \cdot 37^{(\frac{3.5}{5} - \frac{14}{5})} = 21^{\frac{5.0 + 14 - 3.5}{5}} \cdot 37^{\frac{3.5 - 14}{5}} \] At this stage, you would numerically compute the values of \( f(14) \) using a calculator for exact results. Upon performing these calculations, you will find that: \[ f(14) \approx 69.38 \] Thus, the final answer, rounded to the nearest hundredth, is: \[ \boxed{69.38} \]