Answer :
To solve the problem of finding [tex]\( f(10.5) \)[/tex] for the exponential function [tex]\( f(x) = a b^x \)[/tex], given that [tex]\( f(3.5) = 16 \)[/tex] and [tex]\( f(7) = 62 \)[/tex], follow these steps:
1. Understand the Form:
[tex]\( f(x) = a b^x \)[/tex] is the general form of an exponential function, where [tex]\( a \)[/tex] is the initial value and [tex]\( b \)[/tex] is the base, or growth factor.
2. Set Up Two Equations:
Based on the points given:
- [tex]\( f(3.5) = 16 \)[/tex] implies [tex]\( a b^{3.5} = 16 \)[/tex].
- [tex]\( f(7) = 62 \)[/tex] implies [tex]\( a b^7 = 62 \)[/tex].
3. Solve for the Ratio [tex]\( b \)[/tex]:
Divide the second equation by the first to eliminate [tex]\( a \)[/tex] and solve for [tex]\( b \)[/tex]:
[tex]\[
\frac{a b^7}{a b^{3.5}} = \frac{62}{16}
\][/tex]
Simplifies to:
[tex]\[
b^{7-3.5} = \frac{62}{16}
\][/tex]
Calculate the ratio:
[tex]\[
b^{3.5} = \frac{62}{16}
\][/tex]
Take the 3.5 root to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \left(\frac{62}{16}\right)^{\frac{1}{3.5}}
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
Use one of the original equations to find [tex]\( a \)[/tex]. Using [tex]\( f(3.5) = 16 \)[/tex]:
[tex]\[
a = \frac{16}{b^{3.5}}
\][/tex]
5. Find [tex]\( f(10.5) \)[/tex]:
Use the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to find [tex]\( f(10.5) \)[/tex]:
[tex]\[
f(10.5) = a b^{10.5}
\][/tex]
6. Calculate the Final Answer:
Having calculated the necessary values and performed the substitution, we find:
[tex]\[
f(10.5) \approx 240.25
\][/tex]
Thus, the value of [tex]\( f(10.5) \)[/tex], rounded to the nearest hundredth, is approximately [tex]\( 240.25 \)[/tex].
1. Understand the Form:
[tex]\( f(x) = a b^x \)[/tex] is the general form of an exponential function, where [tex]\( a \)[/tex] is the initial value and [tex]\( b \)[/tex] is the base, or growth factor.
2. Set Up Two Equations:
Based on the points given:
- [tex]\( f(3.5) = 16 \)[/tex] implies [tex]\( a b^{3.5} = 16 \)[/tex].
- [tex]\( f(7) = 62 \)[/tex] implies [tex]\( a b^7 = 62 \)[/tex].
3. Solve for the Ratio [tex]\( b \)[/tex]:
Divide the second equation by the first to eliminate [tex]\( a \)[/tex] and solve for [tex]\( b \)[/tex]:
[tex]\[
\frac{a b^7}{a b^{3.5}} = \frac{62}{16}
\][/tex]
Simplifies to:
[tex]\[
b^{7-3.5} = \frac{62}{16}
\][/tex]
Calculate the ratio:
[tex]\[
b^{3.5} = \frac{62}{16}
\][/tex]
Take the 3.5 root to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \left(\frac{62}{16}\right)^{\frac{1}{3.5}}
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
Use one of the original equations to find [tex]\( a \)[/tex]. Using [tex]\( f(3.5) = 16 \)[/tex]:
[tex]\[
a = \frac{16}{b^{3.5}}
\][/tex]
5. Find [tex]\( f(10.5) \)[/tex]:
Use the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to find [tex]\( f(10.5) \)[/tex]:
[tex]\[
f(10.5) = a b^{10.5}
\][/tex]
6. Calculate the Final Answer:
Having calculated the necessary values and performed the substitution, we find:
[tex]\[
f(10.5) \approx 240.25
\][/tex]
Thus, the value of [tex]\( f(10.5) \)[/tex], rounded to the nearest hundredth, is approximately [tex]\( 240.25 \)[/tex].
