Answer :
To solve this problem, we need to determine the exponential function [tex]\( f(x) = a \cdot b^x \)[/tex] using the given points: [tex]\( f(3) = 4 \)[/tex] and [tex]\( f(8.5) = 76 \)[/tex]. After finding the parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we can calculate [tex]\( f(5.5) \)[/tex].
### Step 1: Set Up Equations
1. From the first point, we have:
[tex]\[
a \cdot b^3 = 4
\][/tex]
2. From the second point, we have:
[tex]\[
a \cdot b^{8.5} = 76
\][/tex]
### Step 2: Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]
To eliminate [tex]\( a \)[/tex], divide the second equation by the first equation:
[tex]\[
\frac{a \cdot b^{8.5}}{a \cdot b^3} = \frac{76}{4}
\][/tex]
This simplifies to:
[tex]\[
b^{8.5 - 3} = 19
\][/tex]
[tex]\[
b^{5.5} = 19
\][/tex]
Now, solve for [tex]\( b \)[/tex] by taking the fifth root:
[tex]\[
b = 19^{1/5.5}
\][/tex]
### Step 3: Calculate [tex]\( b \)[/tex]
Using a calculator, approximate:
[tex]\[
b \approx 1.943
\][/tex]
### Step 4: Solve for [tex]\( a \)[/tex]
Substitute [tex]\( b \)[/tex] back into the first equation to find [tex]\( a \)[/tex]:
[tex]\[
a \cdot (1.943)^3 = 4
\][/tex]
Calculate [tex]\( (1.943)^3 \)[/tex]:
[tex]\[
(1.943)^3 \approx 7.324
\][/tex]
[tex]\[
a \cdot 7.324 = 4
\][/tex]
[tex]\[
a = \frac{4}{7.324} \approx 0.546
\][/tex]
### Step 5: Find [tex]\( f(5.5) \)[/tex]
Now that you have [tex]\( a \)[/tex] and [tex]\( b \)[/tex], calculate [tex]\( f(5.5) \)[/tex]:
[tex]\[
f(5.5) = a \cdot b^{5.5}
\][/tex]
[tex]\[
f(5.5) = 0.546 \cdot (1.943)^{5.5}
\][/tex]
Substitute and calculate:
[tex]\[
(1.943)^{5.5} \approx 19
\][/tex]
[tex]\[
f(5.5) = 0.546 \cdot 19 \approx 10.374
\][/tex]
### Conclusion
The value of [tex]\( f(5.5) \)[/tex] is approximately [tex]\( \boxed{10.37} \)[/tex].
### Step 1: Set Up Equations
1. From the first point, we have:
[tex]\[
a \cdot b^3 = 4
\][/tex]
2. From the second point, we have:
[tex]\[
a \cdot b^{8.5} = 76
\][/tex]
### Step 2: Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]
To eliminate [tex]\( a \)[/tex], divide the second equation by the first equation:
[tex]\[
\frac{a \cdot b^{8.5}}{a \cdot b^3} = \frac{76}{4}
\][/tex]
This simplifies to:
[tex]\[
b^{8.5 - 3} = 19
\][/tex]
[tex]\[
b^{5.5} = 19
\][/tex]
Now, solve for [tex]\( b \)[/tex] by taking the fifth root:
[tex]\[
b = 19^{1/5.5}
\][/tex]
### Step 3: Calculate [tex]\( b \)[/tex]
Using a calculator, approximate:
[tex]\[
b \approx 1.943
\][/tex]
### Step 4: Solve for [tex]\( a \)[/tex]
Substitute [tex]\( b \)[/tex] back into the first equation to find [tex]\( a \)[/tex]:
[tex]\[
a \cdot (1.943)^3 = 4
\][/tex]
Calculate [tex]\( (1.943)^3 \)[/tex]:
[tex]\[
(1.943)^3 \approx 7.324
\][/tex]
[tex]\[
a \cdot 7.324 = 4
\][/tex]
[tex]\[
a = \frac{4}{7.324} \approx 0.546
\][/tex]
### Step 5: Find [tex]\( f(5.5) \)[/tex]
Now that you have [tex]\( a \)[/tex] and [tex]\( b \)[/tex], calculate [tex]\( f(5.5) \)[/tex]:
[tex]\[
f(5.5) = a \cdot b^{5.5}
\][/tex]
[tex]\[
f(5.5) = 0.546 \cdot (1.943)^{5.5}
\][/tex]
Substitute and calculate:
[tex]\[
(1.943)^{5.5} \approx 19
\][/tex]
[tex]\[
f(5.5) = 0.546 \cdot 19 \approx 10.374
\][/tex]
### Conclusion
The value of [tex]\( f(5.5) \)[/tex] is approximately [tex]\( \boxed{10.37} \)[/tex].
