Answer :
To find the value of [tex]\( f(16.5) \)[/tex] for the exponential function [tex]\( f(x) = ab^x \)[/tex] given the conditions [tex]\( f(4.5) = 20 \)[/tex] and [tex]\( f(11) = 76 \)[/tex], we need to determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] first. Here’s a step-by-step solution:
### Step 1: Set up the equations.
Given:
1. [tex]\( f(4.5) = ab^{4.5} = 20 \)[/tex]
2. [tex]\( f(11) = ab^{11} = 76 \)[/tex]
### Step 2: Solve the system of equations.
Let's divide the second equation by the first to eliminate [tex]\( a \)[/tex] and solve for [tex]\( b \)[/tex]:
[tex]\[
\frac{ab^{11}}{ab^{4.5}} = \frac{76}{20}
\][/tex]
This simplifies to:
[tex]\[
b^{11 - 4.5} = \frac{76}{20}
\][/tex]
[tex]\[
b^{6.5} = \frac{76}{20}
\][/tex]
[tex]\[
b^{6.5} = 3.8
\][/tex]
To solve for [tex]\( b \)[/tex], we take the 6.5-th root of 3.8:
[tex]\[
b = 3.8^{1/6.5}
\][/tex]
### Step 3: Calculate [tex]\( a \)[/tex].
Now that we have a value for [tex]\( b \)[/tex], we can solve for [tex]\( a \)[/tex] using one of the original equations, say [tex]\( ab^{4.5} = 20 \)[/tex].
[tex]\[
a \cdot (3.8^{1/6.5})^{4.5} = 20
\][/tex]
[tex]\[
a \cdot 3.8^{4.5/6.5} = 20
\][/tex]
[tex]\[
a = \frac{20}{3.8^{4.5/6.5}}
\][/tex]
### Step 4: Calculate [tex]\( f(16.5) \)[/tex].
Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the function to find [tex]\( f(16.5) \)[/tex]:
[tex]\[
f(16.5) = ab^{16.5}
\][/tex]
[tex]\[
f(16.5) = \left(\frac{20}{3.8^{4.5/6.5}}\right) \cdot (3.8^{1/6.5})^{16.5}
\][/tex]
[tex]\[
f(16.5) = \frac{20 \cdot 3.8^{16.5/6.5}}{3.8^{4.5/6.5}}
\][/tex]
Simplify it:
[tex]\[
f(16.5) = 20 \cdot 3.8^{(16.5-4.5)/6.5}
\][/tex]
[tex]\[
f(16.5) = 20 \cdot 3.8^{12/6.5}
\][/tex]
Now we can compute the numerical value using a calculator:
[tex]\[
f(16.5) \approx 20 \cdot 3.8^{1.8461538}
\][/tex]
[tex]\[
f(16.5) \approx 20 \cdot 7.37 \approx 147.4
\][/tex]
Therefore, the value of [tex]\( f(16.5) \)[/tex], rounded to the nearest hundredth, is [tex]\( \boxed{147.40} \)[/tex].
### Step 1: Set up the equations.
Given:
1. [tex]\( f(4.5) = ab^{4.5} = 20 \)[/tex]
2. [tex]\( f(11) = ab^{11} = 76 \)[/tex]
### Step 2: Solve the system of equations.
Let's divide the second equation by the first to eliminate [tex]\( a \)[/tex] and solve for [tex]\( b \)[/tex]:
[tex]\[
\frac{ab^{11}}{ab^{4.5}} = \frac{76}{20}
\][/tex]
This simplifies to:
[tex]\[
b^{11 - 4.5} = \frac{76}{20}
\][/tex]
[tex]\[
b^{6.5} = \frac{76}{20}
\][/tex]
[tex]\[
b^{6.5} = 3.8
\][/tex]
To solve for [tex]\( b \)[/tex], we take the 6.5-th root of 3.8:
[tex]\[
b = 3.8^{1/6.5}
\][/tex]
### Step 3: Calculate [tex]\( a \)[/tex].
Now that we have a value for [tex]\( b \)[/tex], we can solve for [tex]\( a \)[/tex] using one of the original equations, say [tex]\( ab^{4.5} = 20 \)[/tex].
[tex]\[
a \cdot (3.8^{1/6.5})^{4.5} = 20
\][/tex]
[tex]\[
a \cdot 3.8^{4.5/6.5} = 20
\][/tex]
[tex]\[
a = \frac{20}{3.8^{4.5/6.5}}
\][/tex]
### Step 4: Calculate [tex]\( f(16.5) \)[/tex].
Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the function to find [tex]\( f(16.5) \)[/tex]:
[tex]\[
f(16.5) = ab^{16.5}
\][/tex]
[tex]\[
f(16.5) = \left(\frac{20}{3.8^{4.5/6.5}}\right) \cdot (3.8^{1/6.5})^{16.5}
\][/tex]
[tex]\[
f(16.5) = \frac{20 \cdot 3.8^{16.5/6.5}}{3.8^{4.5/6.5}}
\][/tex]
Simplify it:
[tex]\[
f(16.5) = 20 \cdot 3.8^{(16.5-4.5)/6.5}
\][/tex]
[tex]\[
f(16.5) = 20 \cdot 3.8^{12/6.5}
\][/tex]
Now we can compute the numerical value using a calculator:
[tex]\[
f(16.5) \approx 20 \cdot 3.8^{1.8461538}
\][/tex]
[tex]\[
f(16.5) \approx 20 \cdot 7.37 \approx 147.4
\][/tex]
Therefore, the value of [tex]\( f(16.5) \)[/tex], rounded to the nearest hundredth, is [tex]\( \boxed{147.40} \)[/tex].
