Answer :
To solve this problem, we will determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] in the exponential function [tex]\( f(x) = a b^x \)[/tex] using the given conditions, and then we'll use these to calculate [tex]\( f(16.5) \)[/tex].
### Step 1: Set Up Equations
We have two conditions given:
1. [tex]\( f(4.5) = 20 \)[/tex], which gives us the equation:
[tex]\[
a \cdot b^{4.5} = 20
\][/tex]
2. [tex]\( f(11) = 76 \)[/tex], which gives us the equation:
[tex]\[
a \cdot b^{11} = 76
\][/tex]
### Step 2: Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]
To solve these equations, first express [tex]\( a \)[/tex] from one of the equations and substitute it into the other. Let's solve for [tex]\( a \)[/tex] from the first equation:
[tex]\[
a = \frac{20}{b^{4.5}}
\][/tex]
Substitute [tex]\( a \)[/tex] into the second equation:
[tex]\[
\frac{20}{b^{4.5}} \cdot b^{11} = 76
\][/tex]
Simplify this:
[tex]\[
20 \cdot b^{6.5} = 76
\][/tex]
Divide both sides by 20:
[tex]\[
b^{6.5} = \frac{76}{20} = 3.8
\][/tex]
Now solve for [tex]\( b \)[/tex] by taking the [tex]\( 6.5 \)[/tex]-th root:
[tex]\[
b = 3.8^{1/6.5}
\][/tex]
Calculate [tex]\( b \)[/tex]:
Using a calculator, [tex]\( b \approx 1.2108 \)[/tex].
### Step 3: Solve for [tex]\( a \)[/tex]
Substitute [tex]\( b \approx 1.2108 \)[/tex] back into the equation for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{20}{(1.2108)^{4.5}}
\][/tex]
Calculate [tex]\( a \)[/tex]:
Using a calculator, [tex]\( a \approx 9.0748 \)[/tex].
### Step 4: Find [tex]\( f(16.5) \)[/tex]
Now that we have [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we can find [tex]\( f(16.5) \)[/tex]:
[tex]\[
f(16.5) = a \cdot b^{16.5}
\][/tex]
Substitute the values:
[tex]\[
f(16.5) = 9.0748 \cdot (1.2108)^{16.5}
\][/tex]
Calculate [tex]\( f(16.5) \)[/tex]:
Using a calculator, [tex]\( f(16.5) \approx 144.98 \)[/tex].
Therefore, [tex]\( f(16.5) \)[/tex] is approximately [tex]\( 144.98 \)[/tex] to the nearest hundredth.
### Step 1: Set Up Equations
We have two conditions given:
1. [tex]\( f(4.5) = 20 \)[/tex], which gives us the equation:
[tex]\[
a \cdot b^{4.5} = 20
\][/tex]
2. [tex]\( f(11) = 76 \)[/tex], which gives us the equation:
[tex]\[
a \cdot b^{11} = 76
\][/tex]
### Step 2: Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]
To solve these equations, first express [tex]\( a \)[/tex] from one of the equations and substitute it into the other. Let's solve for [tex]\( a \)[/tex] from the first equation:
[tex]\[
a = \frac{20}{b^{4.5}}
\][/tex]
Substitute [tex]\( a \)[/tex] into the second equation:
[tex]\[
\frac{20}{b^{4.5}} \cdot b^{11} = 76
\][/tex]
Simplify this:
[tex]\[
20 \cdot b^{6.5} = 76
\][/tex]
Divide both sides by 20:
[tex]\[
b^{6.5} = \frac{76}{20} = 3.8
\][/tex]
Now solve for [tex]\( b \)[/tex] by taking the [tex]\( 6.5 \)[/tex]-th root:
[tex]\[
b = 3.8^{1/6.5}
\][/tex]
Calculate [tex]\( b \)[/tex]:
Using a calculator, [tex]\( b \approx 1.2108 \)[/tex].
### Step 3: Solve for [tex]\( a \)[/tex]
Substitute [tex]\( b \approx 1.2108 \)[/tex] back into the equation for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{20}{(1.2108)^{4.5}}
\][/tex]
Calculate [tex]\( a \)[/tex]:
Using a calculator, [tex]\( a \approx 9.0748 \)[/tex].
### Step 4: Find [tex]\( f(16.5) \)[/tex]
Now that we have [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we can find [tex]\( f(16.5) \)[/tex]:
[tex]\[
f(16.5) = a \cdot b^{16.5}
\][/tex]
Substitute the values:
[tex]\[
f(16.5) = 9.0748 \cdot (1.2108)^{16.5}
\][/tex]
Calculate [tex]\( f(16.5) \)[/tex]:
Using a calculator, [tex]\( f(16.5) \approx 144.98 \)[/tex].
Therefore, [tex]\( f(16.5) \)[/tex] is approximately [tex]\( 144.98 \)[/tex] to the nearest hundredth.