Answer :
Sure! Let's find the value of [tex]\( f(16.5) \)[/tex] for the exponential function [tex]\( f(x) = a \cdot b^x \)[/tex], given [tex]\( f(4.5) = 20 \)[/tex] and [tex]\( f(11) = 76 \)[/tex]. Here is how you can approach this problem step-by-step:
1. Understand the Function's Form: We have an exponential function [tex]\( f(x) = a \cdot b^x \)[/tex].
2. Set Up the Equations: Using the given points, we can write two equations:
- [tex]\( a \cdot b^{4.5} = 20 \)[/tex]
- [tex]\( a \cdot b^{11} = 76 \)[/tex]
3. Solve for [tex]\( b \)[/tex]: Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^{11}}{a \cdot b^{4.5}} = \frac{76}{20}
\][/tex]
Simplifying gives us:
[tex]\[
b^{11 - 4.5} = \frac{76}{20}
\][/tex]
[tex]\[
b^{6.5} = 3.8
\][/tex]
Solve for [tex]\( b \)[/tex] by taking the 6.5th root:
[tex]\[
b \approx 1.228
\][/tex]
4. Find [tex]\( a \)[/tex]: Substitute the value of [tex]\( b \)[/tex] back into one of the original equations to solve for [tex]\( a \)[/tex]. We'll use [tex]\( a \cdot b^{4.5} = 20 \)[/tex]:
[tex]\[
a \cdot (1.228)^{4.5} = 20
\][/tex]
[tex]\[
a \approx 7.937
\][/tex]
5. Calculate [tex]\( f(16.5) \)[/tex]: Now substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the function and calculate [tex]\( f(16.5) \)[/tex]:
[tex]\[
f(16.5) = a \cdot b^{16.5}
\][/tex]
[tex]\[
f(16.5) \approx 7.937 \cdot (1.228)^{16.5}
\][/tex]
[tex]\[
f(16.5) \approx 235.18
\][/tex]
So, the value of [tex]\( f(16.5) \)[/tex] is approximately 235.18.
1. Understand the Function's Form: We have an exponential function [tex]\( f(x) = a \cdot b^x \)[/tex].
2. Set Up the Equations: Using the given points, we can write two equations:
- [tex]\( a \cdot b^{4.5} = 20 \)[/tex]
- [tex]\( a \cdot b^{11} = 76 \)[/tex]
3. Solve for [tex]\( b \)[/tex]: Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^{11}}{a \cdot b^{4.5}} = \frac{76}{20}
\][/tex]
Simplifying gives us:
[tex]\[
b^{11 - 4.5} = \frac{76}{20}
\][/tex]
[tex]\[
b^{6.5} = 3.8
\][/tex]
Solve for [tex]\( b \)[/tex] by taking the 6.5th root:
[tex]\[
b \approx 1.228
\][/tex]
4. Find [tex]\( a \)[/tex]: Substitute the value of [tex]\( b \)[/tex] back into one of the original equations to solve for [tex]\( a \)[/tex]. We'll use [tex]\( a \cdot b^{4.5} = 20 \)[/tex]:
[tex]\[
a \cdot (1.228)^{4.5} = 20
\][/tex]
[tex]\[
a \approx 7.937
\][/tex]
5. Calculate [tex]\( f(16.5) \)[/tex]: Now substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the function and calculate [tex]\( f(16.5) \)[/tex]:
[tex]\[
f(16.5) = a \cdot b^{16.5}
\][/tex]
[tex]\[
f(16.5) \approx 7.937 \cdot (1.228)^{16.5}
\][/tex]
[tex]\[
f(16.5) \approx 235.18
\][/tex]
So, the value of [tex]\( f(16.5) \)[/tex] is approximately 235.18.
