Answer :
To solve the problem, we need to determine the value of the exponential function [tex]\( f(x) = a \cdot b^x \)[/tex] at [tex]\( x = 1.5 \)[/tex], given that [tex]\( f(0.5) = 26 \)[/tex] and [tex]\( f(1) = 66 \)[/tex].
### Step 1: Set up the equations from given values
We have two points on the function:
- [tex]\( f(0.5) = a \cdot b^{0.5} = 26 \)[/tex]
- [tex]\( f(1) = a \cdot b^1 = 66 \)[/tex]
### Step 2: Eliminate 'a' to solve for 'b'
Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b}{a \cdot b^{0.5}} = \frac{66}{26}
\][/tex]
[tex]\[
b^{1 - 0.5} = \frac{66}{26}
\][/tex]
[tex]\[
b^{0.5} = \frac{66}{26} \approx 2.538461538
\][/tex]
### Step 3: Solve for 'b' by squaring
Square both sides to find [tex]\( b \)[/tex]:
[tex]\[
b = (2.538461538)^2 \approx 6.44
\][/tex]
### Step 4: Substitute 'b' back to find 'a'
Use the first equation to solve for [tex]\( a \)[/tex]:
[tex]\[
a \cdot 6.44^{0.5} = 26
\][/tex]
[tex]\[
a \cdot 2.538461538 = 26
\][/tex]
[tex]\[
a = \frac{26}{2.538461538} \approx 10.24
\][/tex]
### Step 5: Calculate [tex]\( f(1.5) \)[/tex]
Now use the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to find [tex]\( f(1.5) \)[/tex]:
[tex]\[
f(1.5) = a \cdot b^{1.5}
\][/tex]
[tex]\[
f(1.5) = 10.24 \cdot 6.44^{1.5} \approx 167.54
\][/tex]
Therefore, the value of [tex]\( f(1.5) \)[/tex], rounded to the nearest hundredth, is [tex]\( \boxed{167.54} \)[/tex].
### Step 1: Set up the equations from given values
We have two points on the function:
- [tex]\( f(0.5) = a \cdot b^{0.5} = 26 \)[/tex]
- [tex]\( f(1) = a \cdot b^1 = 66 \)[/tex]
### Step 2: Eliminate 'a' to solve for 'b'
Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b}{a \cdot b^{0.5}} = \frac{66}{26}
\][/tex]
[tex]\[
b^{1 - 0.5} = \frac{66}{26}
\][/tex]
[tex]\[
b^{0.5} = \frac{66}{26} \approx 2.538461538
\][/tex]
### Step 3: Solve for 'b' by squaring
Square both sides to find [tex]\( b \)[/tex]:
[tex]\[
b = (2.538461538)^2 \approx 6.44
\][/tex]
### Step 4: Substitute 'b' back to find 'a'
Use the first equation to solve for [tex]\( a \)[/tex]:
[tex]\[
a \cdot 6.44^{0.5} = 26
\][/tex]
[tex]\[
a \cdot 2.538461538 = 26
\][/tex]
[tex]\[
a = \frac{26}{2.538461538} \approx 10.24
\][/tex]
### Step 5: Calculate [tex]\( f(1.5) \)[/tex]
Now use the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to find [tex]\( f(1.5) \)[/tex]:
[tex]\[
f(1.5) = a \cdot b^{1.5}
\][/tex]
[tex]\[
f(1.5) = 10.24 \cdot 6.44^{1.5} \approx 167.54
\][/tex]
Therefore, the value of [tex]\( f(1.5) \)[/tex], rounded to the nearest hundredth, is [tex]\( \boxed{167.54} \)[/tex].
