Answer :
To solve this problem, we have an exponential function of the form [tex]\( f(x) = a \cdot b^x \)[/tex]. We're given two conditions: [tex]\( f(3) = 25 \)[/tex] and [tex]\( f(5) = 83 \)[/tex]. We'll use these conditions to find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], and then calculate [tex]\( f(4.5) \)[/tex].
Step 1: Set up the equations
From the given function form, we have two equations:
1. [tex]\( a \cdot b^3 = 25 \)[/tex]
2. [tex]\( a \cdot b^5 = 83 \)[/tex]
Step 2: Divide the equations to find [tex]\( b \)[/tex]
Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^5}{a \cdot b^3} = \frac{83}{25}
\][/tex]
This simplifies to:
[tex]\[
b^2 = \frac{83}{25}
\][/tex]
Calculate [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 \approx 3.32
\][/tex]
Now, find [tex]\( b \)[/tex] by taking the square root:
[tex]\[
b \approx 1.8221
\][/tex]
Step 3: Solve for [tex]\( a \)[/tex]
Substitute the value of [tex]\( b \)[/tex] back into one of the original equations, like [tex]\( a \cdot b^3 = 25 \)[/tex]:
[tex]\[
a \cdot (1.8221)^3 = 25
\][/tex]
Calculate [tex]\( a \)[/tex]:
[tex]\[
a \approx \frac{25}{6.0403} \approx 4.1327
\][/tex]
Step 4: Calculate [tex]\( f(4.5) \)[/tex]
Now that we have [tex]\( a \)[/tex] and [tex]\( b \)[/tex], use these to find [tex]\( f(4.5) \)[/tex]:
[tex]\[
f(4.5) = a \cdot b^{4.5}
\][/tex]
[tex]\[
f(4.5) = 4.1327 \cdot (1.8221)^{4.5}
\][/tex]
[tex]\[
f(4.5) \approx 61.49
\][/tex]
So, the value of [tex]\( f(4.5) \)[/tex], to the nearest hundredth, is approximately 61.49.
Step 1: Set up the equations
From the given function form, we have two equations:
1. [tex]\( a \cdot b^3 = 25 \)[/tex]
2. [tex]\( a \cdot b^5 = 83 \)[/tex]
Step 2: Divide the equations to find [tex]\( b \)[/tex]
Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^5}{a \cdot b^3} = \frac{83}{25}
\][/tex]
This simplifies to:
[tex]\[
b^2 = \frac{83}{25}
\][/tex]
Calculate [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 \approx 3.32
\][/tex]
Now, find [tex]\( b \)[/tex] by taking the square root:
[tex]\[
b \approx 1.8221
\][/tex]
Step 3: Solve for [tex]\( a \)[/tex]
Substitute the value of [tex]\( b \)[/tex] back into one of the original equations, like [tex]\( a \cdot b^3 = 25 \)[/tex]:
[tex]\[
a \cdot (1.8221)^3 = 25
\][/tex]
Calculate [tex]\( a \)[/tex]:
[tex]\[
a \approx \frac{25}{6.0403} \approx 4.1327
\][/tex]
Step 4: Calculate [tex]\( f(4.5) \)[/tex]
Now that we have [tex]\( a \)[/tex] and [tex]\( b \)[/tex], use these to find [tex]\( f(4.5) \)[/tex]:
[tex]\[
f(4.5) = a \cdot b^{4.5}
\][/tex]
[tex]\[
f(4.5) = 4.1327 \cdot (1.8221)^{4.5}
\][/tex]
[tex]\[
f(4.5) \approx 61.49
\][/tex]
So, the value of [tex]\( f(4.5) \)[/tex], to the nearest hundredth, is approximately 61.49.
