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------------------------------------------------ If [tex]f(x)[/tex] is an exponential function of the form [tex]y = ab^x[/tex], where [tex]f(4.5) = 10[/tex] and [tex]f(8.5) = 66[/tex], then find the value of [tex]f(14.5)[/tex] to the nearest hundredth.

To solve the problem of finding [tex]\( f(14.5) \)[/tex] for the exponential function [tex]\( y = a \cdot b^x \)[/tex] given [tex]\( f(4.5) = 10 \)[/tex] and [tex]\( f(8.5) = 66 \)[/tex], follow these steps:

1. Set up the equations using the given points:
- From [tex]\( f(4.5) = 10 \)[/tex], we have the equation:
[tex]\( a \cdot b^{4.5} = 10 \)[/tex].
- From [tex]\( f(8.5) = 66 \)[/tex], we have the equation:
[tex]\( a \cdot b^{8.5} = 66 \)[/tex].

2. Solve for [tex]\( b \)[/tex]:
- Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^{8.5}}{a \cdot b^{4.5}} = \frac{66}{10}
\][/tex]
- Simplify to find:
[tex]\[
b^{8.5 - 4.5} = 6.6
\][/tex]
[tex]\[
b^4 = 6.6
\][/tex]
- Take the fourth root to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \sqrt[4]{6.6}
\][/tex]
- In numerical terms, [tex]\( b \approx 1.60 \)[/tex].

3. Solve for [tex]\( a \)[/tex]:
- Use the first equation:
[tex]\( a \cdot b^{4.5} = 10 \)[/tex].
- Substitute the value of [tex]\( b \)[/tex]:
[tex]\[
a \cdot (1.60)^{4.5} = 10
\][/tex]
- Solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{10}{(1.60)^{4.5}}
\][/tex]
- Numerically, [tex]\( a \approx 1.20 \)[/tex].

4. Find [tex]\( f(14.5) \)[/tex]:
- Use the formula [tex]\( f(x) = a \cdot b^x \)[/tex] with [tex]\( x = 14.5 \)[/tex]:
[tex]\[
f(14.5) = 1.20 \cdot (1.60)^{14.5}
\][/tex]
- Calculate the value:
[tex]\[
f(14.5) \approx 1119.08
\][/tex]

So, the value of [tex]\( f(14.5) \)[/tex] is approximately 1119.08 to the nearest hundredth.