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.

Given that \( f(x) \) is an exponential function of the form \( y = ab^x \), where \( f(4.5) = 20 \) and \( f(11) = 76 \), we need to find the value of \( f(16.5) \) to the nearest hundredth. To find the value of \( f(16.5) \), we first need to determine the values of \( a \) and \( b \) using the given information. We have two equations: 1. \( 20 = ab^{4.5} \) 2. \( 76 = ab^{11} \) We can solve these equations simultaneously to find the values of \( a \) and \( b \). Let's solve the system of equations to find the values of \( a \) and \( b \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}20=ab^{4.5}\\76=ab^{11}\end{array}\right.\) - step1: Calculate: \(\left\{ \begin{array}{l}20=ab^{\frac{9}{2}}\\76=ab^{11}\end{array}\right.\) - step2: Solve the equation: \(\left\{ \begin{array}{l}a=\frac{20\sqrt{b}}{b^{5}}\\76=ab^{11}\end{array}\right.\) - step3: Substitute the value of \(a:\) \(76=\frac{20\sqrt{b}}{b^{5}}\times b^{11}\) - step4: Reduce the fraction: \(76=20\sqrt{b}\times b^{6}\) - step5: Swap the sides: \(20\sqrt{b}\times b^{6}=76\) - step6: Move the expression to the left side: \(20\sqrt{b}\times b^{6}-76=0\) - step7: Evaluate: \(20b^{6}\sqrt{b}-76=0\) - step8: Move the constant to the right-hand side: \(20b^{6}\sqrt{b}=76\) - step9: Divide both sides: \(5b^{6}\sqrt{b}=19\) - step10: Raise both sides to the \(2\)-th power\(:\) \(\left(5b^{6}\sqrt{b}\right)^{2}=19^{2}\) - step11: Evaluate the power: \(25b^{13}=361\) - step12: Divide both sides: \(\frac{25b^{13}}{25}=\frac{361}{25}\) - step13: Divide the numbers: \(b^{13}=\frac{361}{25}\) - step14: Simplify the equation: \(\sqrt[13]{b^{13}}=\sqrt[13]{\frac{361}{25}}\) - step15: Calculate: \(b=\sqrt[13]{\frac{361}{25}}\) - step16: Simplify the root: \(b=\frac{\sqrt[13]{361\times 5^{11}}}{5}\) - step17: Substitute the value of \(b:\) \(a=\frac{20\sqrt{\frac{\sqrt[13]{361\times 5^{11}}}{5}}}{\left(\frac{\sqrt[13]{361\times 5^{11}}}{5}\right)^{5}}\) - step18: Simplify: \(a=\frac{4\sqrt[13]{19\times 5^{12}\left(361\times 5^{11}\right)^{8}}}{5640625}\) - step19: Calculate: \(a=\frac{4\sqrt[13]{19^{17}\times 5^{100}}}{5640625}\) - step20: Calculate: \(\left\{ \begin{array}{l}a=\frac{4\sqrt[13]{19^{17}\times 5^{100}}}{5640625}\\b=\frac{\sqrt[13]{361\times 5^{11}}}{5}\end{array}\right.\) - step21: Check the solution: \(\left\{ \begin{array}{l}a=\frac{4\sqrt[13]{19^{17}\times 5^{100}}}{5640625}\\b=\frac{\sqrt[13]{361\times 5^{11}}}{5}\end{array}\right.\) - step22: Rewrite: \(\left(a,b\right) = \left(\frac{4\sqrt[13]{19^{17}\times 5^{100}}}{5640625},\frac{\sqrt[13]{361\times 5^{11}}}{5}\right)\) The solution to the system of equations is \( a = \frac{4\sqrt[13]{19^{17} \times 5^{100}}}{5640625} \) and \( b = \frac{\sqrt[13]{361 \times 5^{11}}}{5} \). Now that we have the values of \( a \) and \( b \), we can substitute them into the exponential function \( f(x) = ab^x \) to find the value of \( f(16.5) \). Substitute \( a = \frac{4\sqrt[13]{19^{17} \times 5^{100}}}{5640625} \) and \( b = \frac{\sqrt[13]{361 \times 5^{11}}}{5} \) into the function \( f(x) = ab^x \) to find \( f(16.5) \). Calculate the value by following steps: - step0: Calculate: \(\frac{4\sqrt[13]{19^{17}\times 5^{100}}}{5640625}\left(\frac{\sqrt[13]{361\times 5^{11}}}{5}\right)^{16.5}\) - step1: Convert the expressions: \(\frac{4\sqrt[13]{19^{17}\times 5^{100}}}{5640625}\left(\frac{\sqrt[13]{361\times 5^{11}}}{5}\right)^{\frac{33}{2}}\) - step2: Evaluate the power: \(\frac{4\sqrt[13]{19^{17}\times 5^{100}}}{5640625}\times \frac{\left(\sqrt[13]{361\times 5^{11}}\right)^{\frac{33}{2}}}{5^{\frac{33}{2}}}\) - step3: Multiply the fractions: \(\frac{4\sqrt[13]{19^{17}\times 5^{100}}\times \left(\sqrt[13]{361\times 5^{11}}\right)^{\frac{33}{2}}}{5640625\times 5^{\frac{33}{2}}}\) - step4: Multiply: \(\frac{4\left(\sqrt[13]{361\times 5^{11}}\right)^{\frac{33}{2}}\sqrt[13]{19^{17}\times 5^{100}}}{5640625\times 5^{\frac{33}{2}}}\) - step5: Calculate: \(\frac{1444\times 5^{11}\sqrt[26]{\left(361\times 5^{11}\right)^{7}\left(19^{17}\times 5^{100}\right)^{2}}}{5640625\times 5^{\frac{33}{2}}}\) - step6: Reduce the fraction: \(\frac{4\sqrt[26]{\left(361\times 5^{11}\right)^{7}\left(19^{17}\times 5^{100}\right)^{2}}}{5^{\frac{23}{2}}}\) - step7: Simplify: \(\frac{4\sqrt[26]{\left(361\times 5^{11}\right)^{7}\left(19^{17}\times 5^{100}\right)^{2}}}{5^{11}\sqrt{5}}\) - step8: Multiply by the Conjugate: \(\frac{4\sqrt[26]{\left(361\times 5^{11}\right)^{7}\left(19^{17}\times 5^{100}\right)^{2}}\times \sqrt{5}}{5^{11}\sqrt{5}\times \sqrt{5}}\) - step9: Multiply the numbers: \(\frac{4\sqrt[26]{\left(361\times 5^{11}\right)^{7}\left(19^{17}\times 5^{100}\right)^{2}\times 5^{13}}}{5^{11}\sqrt{5}\times \sqrt{5}}\) - step10: Multiply the numbers: \(\frac{4\sqrt[26]{\left(361\times 5^{11}\right)^{7}\left(19^{17}\times 5^{100}\right)^{2}\times 5^{13}}}{5^{12}}\) - step11: Expand the expression: \(\frac{4\sqrt[26]{19^{48}\times 5^{290}}}{5^{12}}\) The value of \( f(16.5) \) to the nearest hundredth is approximately 235.18.