If \( f(x) \) is an exponential function of the form of \( y=a b^{x} \) where \( f(4.5)=20 \) and \( f(11)=76 \), then find the value of \( f(16.5) \), to the nearest hundredth.

To solve for \( f(16.5) \) in the exponential function \( f(x) = a b^x \) given the values \( f(4.5) = 20 \) and \( f(11) = 76 \), we can set up two equations based on those values. 1. From the first point: \[ 20 = a b^{4.5} \quad \text{(1)} \] 2. From the second point: \[ 76 = a b^{11} \quad \text{(2)} \] Next, we can divide equation (2) by equation (1) to eliminate \( a \): \[ \frac{76}{20} = \frac{a b^{11}}{a b^{4.5}} \] This simplifies to: \[ 3.8 = b^{11-4.5} = b^{6.5} \] Now we take both sides to the power of \( \frac{1}{6.5} \): \[ b = 3.8^{\frac{1}{6.5}} \] Calculating \( b \): \[ b \approx 1.66 \] Now we can use this value of \( b \) to find \( a \). We can substitute back into equation (1): \[ 20 = a (1.66)^{4.5} \] Calculating \( (1.66)^{4.5} \): \[ (1.66)^{4.5} \approx 6.83 \] So we have: \[ 20 = a \cdot 6.83 \] Therefore: \[ a \approx \frac{20}{6.83} \approx 2.93 \] Now that we have values for \( a \) and \( b \): \[ a \approx 2.93, \quad b \approx 1.66 \] Now we can find \( f(16.5) \): \[ f(16.5) = a b^{16.5} \approx 2.93 \cdot (1.66)^{16.5} \] Calculating \( (1.66)^{16.5} \): \[ (1.66)^{16.5} \approx 113.3 \] Finally, we calculate: \[ f(16.5) \approx 2.93 \cdot 113.3 \approx 332.86 \] Thus, the value of \( f(16.5) \) to the nearest hundredth is: \[ \boxed{332.86} \]