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If f(x) is an exponential function where f(3.5)=12, and f(9)=75 then find the value of f(7.5) to the nearest hundredth?

User Ozplc
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1 Answer

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Given the general exponential function:


f(x)=A\cdot b^x

Where A and b are parameters. Then, using the information provided in the problem:


\begin{gathered} f(3.5)=A\cdot b^(3.5)=12 \\ f(9)=A\cdot b^9=75 \end{gathered}

Taking the division:


\begin{gathered} (A\cdot b^9)/(A\cdot b^(3.5))=(75)/(12) \\ b^(5.5)=6.25 \\ \Rightarrow b=1.39542 \end{gathered}

Now, we use f(9) = 75 to find the parameter A:


\begin{gathered} f(9)=A\cdot1.39542^9=75 \\ \Rightarrow A=3.7386 \end{gathered}

The function is:


f(x)=3.7386\cdot1.39542^x

We evaluate at x = 7.5:


\begin{gathered} f(7.5)=3.7386\cdot1.39542^(7.5) \\ \Rightarrow f(7.5)=45.50 \end{gathered}

User Kirtan
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