Answer :
To solve the problem where [tex]\( f(x) = ab^x \)[/tex] is an exponential function, and you're given [tex]\( f(4.5) = 10 \)[/tex] and [tex]\( f(8.5) = 66 \)[/tex], and you need to find [tex]\( f(14.5) \)[/tex], follow these steps:
1. Set Up the Equations:
- From the information, we know:
[tex]\[
ab^{4.5} = 10
\][/tex]
[tex]\[
ab^{8.5} = 66
\][/tex]
2. Divide the Equations:
- Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{ab^{8.5}}{ab^{4.5}} = \frac{66}{10}
\][/tex]
- Simplify this to:
[tex]\[
b^{8.5 - 4.5} = 6.6
\][/tex]
[tex]\[
b^4 = 6.6
\][/tex]
3. Solve for [tex]\( b \)[/tex]:
- To find [tex]\( b \)[/tex], take the fourth root of 6.6:
[tex]\[
b = 6.6^{1/4}
\][/tex]
- This gives us [tex]\( b \approx 1.6028 \)[/tex].
4. Solve for [tex]\( a \)[/tex]:
- Use the first equation to solve for [tex]\( a \)[/tex] with the value of [tex]\( b \)[/tex]:
[tex]\[
10 = a \times (1.6028)^{4.5}
\][/tex]
- Isolate [tex]\( a \)[/tex]:
[tex]\[
a = \frac{10}{(1.6028)^{4.5}}
\][/tex]
- This gives us [tex]\( a \approx 1.1968 \)[/tex].
5. Find [tex]\( f(14.5) \)[/tex]:
- Plug [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the equation [tex]\( f(x) = ab^x \)[/tex] to find [tex]\( f(14.5) \)[/tex]:
[tex]\[
f(14.5) = 1.1968 \times (1.6028)^{14.5}
\][/tex]
- Calculate this to get [tex]\( f(14.5) \approx 1119.08 \)[/tex].
Therefore, the value of [tex]\( f(14.5) \)[/tex] is approximately 1119.08, rounded to the nearest hundredth.
1. Set Up the Equations:
- From the information, we know:
[tex]\[
ab^{4.5} = 10
\][/tex]
[tex]\[
ab^{8.5} = 66
\][/tex]
2. Divide the Equations:
- Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{ab^{8.5}}{ab^{4.5}} = \frac{66}{10}
\][/tex]
- Simplify this to:
[tex]\[
b^{8.5 - 4.5} = 6.6
\][/tex]
[tex]\[
b^4 = 6.6
\][/tex]
3. Solve for [tex]\( b \)[/tex]:
- To find [tex]\( b \)[/tex], take the fourth root of 6.6:
[tex]\[
b = 6.6^{1/4}
\][/tex]
- This gives us [tex]\( b \approx 1.6028 \)[/tex].
4. Solve for [tex]\( a \)[/tex]:
- Use the first equation to solve for [tex]\( a \)[/tex] with the value of [tex]\( b \)[/tex]:
[tex]\[
10 = a \times (1.6028)^{4.5}
\][/tex]
- Isolate [tex]\( a \)[/tex]:
[tex]\[
a = \frac{10}{(1.6028)^{4.5}}
\][/tex]
- This gives us [tex]\( a \approx 1.1968 \)[/tex].
5. Find [tex]\( f(14.5) \)[/tex]:
- Plug [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the equation [tex]\( f(x) = ab^x \)[/tex] to find [tex]\( f(14.5) \)[/tex]:
[tex]\[
f(14.5) = 1.1968 \times (1.6028)^{14.5}
\][/tex]
- Calculate this to get [tex]\( f(14.5) \approx 1119.08 \)[/tex].
Therefore, the value of [tex]\( f(14.5) \)[/tex] is approximately 1119.08, rounded to the nearest hundredth.
