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], find the value of [tex]f(14.5)[/tex], to the nearest hundredth.

To solve this problem, we need to determine an exponential function of the form [tex]\( y = a \cdot b^x \)[/tex] that passes through the given points [tex]\( f(4.5) = 10 \)[/tex] and [tex]\( f(8.5) = 66 \)[/tex]. Once we find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we can use them to find [tex]\( f(14.5) \)[/tex].

### Step-by-Step Solution:

1. Set Up the Equations:
We know that:
- [tex]\( f(4.5) = 10 \)[/tex]
- [tex]\( f(8.5) = 66 \)[/tex]

These can be written as:
- [tex]\( 10 = a \cdot b^{4.5} \)[/tex]
- [tex]\( 66 = a \cdot b^{8.5} \)[/tex]

2. Create the Ratio:
To eliminate [tex]\( a \)[/tex], we take the ratio of the two equations:
[tex]\[
\frac{66}{10} = \frac{a \cdot b^{8.5}}{a \cdot b^{4.5}}
\][/tex]
Simplifying this gives:
[tex]\[
6.6 = b^{8.5 - 4.5}
\][/tex]
[tex]\[
6.6 = b^4
\][/tex]

3. Solve for [tex]\( b \)[/tex]:
Take the fourth root of both sides to solve for [tex]\( b \)[/tex]:
[tex]\[
b = 6.6^{\frac{1}{4}}
\][/tex]
After evaluating, we find:
[tex]\[
b \approx 1.6028
\][/tex]

4. Solve for [tex]\( a \)[/tex]:
Substitute [tex]\( b \)[/tex] back into one of the original equations to solve for [tex]\( a \)[/tex]. Using [tex]\( 10 = a \cdot b^{4.5} \)[/tex]:
[tex]\[
10 = a \cdot (1.6028)^{4.5}
\][/tex]
[tex]\[
a = \frac{10}{(1.6028)^{4.5}}
\][/tex]
After evaluating, we find:
[tex]\[
a \approx 1.1968
\][/tex]

5. Determine [tex]\( f(14.5) \)[/tex]:
Now, use the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to find [tex]\( f(14.5) \)[/tex]:
[tex]\[
f(14.5) = a \cdot b^{14.5}
\][/tex]
Substitute the values:
[tex]\[
f(14.5) \approx 1.1968 \cdot (1.6028)^{14.5}
\][/tex]
Evaluating this gives:
[tex]\[
f(14.5) \approx 1119.08
\][/tex]

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