Answer :
To find the value of [tex]\( f(1.5) \)[/tex] for the exponential function [tex]\( y = ab^x \)[/tex], we need to determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] using the given points. We know:
1. [tex]\( f(0.5) = 26 \)[/tex] which gives us the equation [tex]\( ab^{0.5} = 26 \)[/tex].
2. [tex]\( f(1) = 66 \)[/tex] which provides the equation [tex]\( ab^1 = 66 \)[/tex].
### Step-by-step Solution:
1. Solve for [tex]\( b \)[/tex]:
From the second equation, we have:
[tex]\[
ab = 66
\][/tex]
Solving for [tex]\( a \)[/tex] gives:
[tex]\[
a = \frac{66}{b}
\][/tex]
2. Substitute [tex]\( a \)[/tex] into the first equation:
Substitute [tex]\( a = \frac{66}{b} \)[/tex] into the first equation [tex]\( ab^{0.5} = 26 \)[/tex]:
[tex]\[
\left(\frac{66}{b}\right) b^{0.5} = 26
\][/tex]
Simplifying this, we have:
[tex]\[
66b^{-0.5} \cdot b^{0.5} = 26 \quad \Rightarrow \quad 66 b^{0} = 26 \quad \Rightarrow \quad 66 = 26b^{0.5}
\][/tex]
Simplify the equation:
[tex]\[
b^{0.5} = \frac{66}{26}
\][/tex]
[tex]\[
b^{0.5} = \frac{33}{13}
\][/tex]
3. Solve for [tex]\( b \)[/tex]:
Squaring both sides to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \left(\frac{33}{13}\right)^2
\][/tex]
[tex]\[
b = \frac{1089}{169}
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
Substitute [tex]\( b = \frac{1089}{169} \)[/tex] back into [tex]\( ab = 66 \)[/tex]:
[tex]\[
a \cdot \frac{1089}{169} = 66
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = 66 \times \frac{169}{1089}
\][/tex]
[tex]\[
a = \frac{11154}{1089}
\][/tex]
[tex]\[
a = \frac{68}{13}
\][/tex]
5. Calculate [tex]\( f(1.5) \)[/tex]:
Now, we find [tex]\( f(1.5) \)[/tex]:
[tex]\[
f(1.5) = ab^{1.5}
\][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(1.5) = \frac{68}{13} \cdot \left(\frac{1089}{169}\right)^{1.5}
\][/tex]
Calculate:
[tex]\[
\sqrt{\frac{1089}{169}} = \frac{33}{13}
\][/tex]
[tex]\[
\left(\frac{1089}{169}\right)^{1.5} = \left(\frac{33}{13}\right)^3 = \frac{35937}{2197}
\][/tex]
Multiply the values:
[tex]\[
f(1.5) = \frac{68}{13} \times \frac{35937}{2197}
\][/tex]
Simplify:
[tex]\[
f(1.5) = \frac{2443716}{28561}
\][/tex]
This results in approximately:
[tex]\[
f(1.5) \approx 168.53
\][/tex]
Therefore, the value of [tex]\( f(1.5) \)[/tex] is approximately [tex]\( 168.53 \)[/tex].
1. [tex]\( f(0.5) = 26 \)[/tex] which gives us the equation [tex]\( ab^{0.5} = 26 \)[/tex].
2. [tex]\( f(1) = 66 \)[/tex] which provides the equation [tex]\( ab^1 = 66 \)[/tex].
### Step-by-step Solution:
1. Solve for [tex]\( b \)[/tex]:
From the second equation, we have:
[tex]\[
ab = 66
\][/tex]
Solving for [tex]\( a \)[/tex] gives:
[tex]\[
a = \frac{66}{b}
\][/tex]
2. Substitute [tex]\( a \)[/tex] into the first equation:
Substitute [tex]\( a = \frac{66}{b} \)[/tex] into the first equation [tex]\( ab^{0.5} = 26 \)[/tex]:
[tex]\[
\left(\frac{66}{b}\right) b^{0.5} = 26
\][/tex]
Simplifying this, we have:
[tex]\[
66b^{-0.5} \cdot b^{0.5} = 26 \quad \Rightarrow \quad 66 b^{0} = 26 \quad \Rightarrow \quad 66 = 26b^{0.5}
\][/tex]
Simplify the equation:
[tex]\[
b^{0.5} = \frac{66}{26}
\][/tex]
[tex]\[
b^{0.5} = \frac{33}{13}
\][/tex]
3. Solve for [tex]\( b \)[/tex]:
Squaring both sides to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \left(\frac{33}{13}\right)^2
\][/tex]
[tex]\[
b = \frac{1089}{169}
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
Substitute [tex]\( b = \frac{1089}{169} \)[/tex] back into [tex]\( ab = 66 \)[/tex]:
[tex]\[
a \cdot \frac{1089}{169} = 66
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = 66 \times \frac{169}{1089}
\][/tex]
[tex]\[
a = \frac{11154}{1089}
\][/tex]
[tex]\[
a = \frac{68}{13}
\][/tex]
5. Calculate [tex]\( f(1.5) \)[/tex]:
Now, we find [tex]\( f(1.5) \)[/tex]:
[tex]\[
f(1.5) = ab^{1.5}
\][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(1.5) = \frac{68}{13} \cdot \left(\frac{1089}{169}\right)^{1.5}
\][/tex]
Calculate:
[tex]\[
\sqrt{\frac{1089}{169}} = \frac{33}{13}
\][/tex]
[tex]\[
\left(\frac{1089}{169}\right)^{1.5} = \left(\frac{33}{13}\right)^3 = \frac{35937}{2197}
\][/tex]
Multiply the values:
[tex]\[
f(1.5) = \frac{68}{13} \times \frac{35937}{2197}
\][/tex]
Simplify:
[tex]\[
f(1.5) = \frac{2443716}{28561}
\][/tex]
This results in approximately:
[tex]\[
f(1.5) \approx 168.53
\][/tex]
Therefore, the value of [tex]\( f(1.5) \)[/tex] is approximately [tex]\( 168.53 \)[/tex].
