Answer :
To solve the problem, we are given an exponential function of the form [tex]\( f(x) = a \cdot b^x \)[/tex] and two specific points: [tex]\( f(3.5) = 16 \)[/tex] and [tex]\( f(7) = 62 \)[/tex]. We need to find [tex]\( f(10.5) \)[/tex].
1. Set up the equations using the given points:
- For [tex]\( f(3.5) = 16 \)[/tex]:
[tex]\[
a \cdot b^{3.5} = 16
\][/tex]
- For [tex]\( f(7) = 62 \)[/tex]:
[tex]\[
a \cdot b^7 = 62
\][/tex]
2. Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^7}{a \cdot b^{3.5}} = \frac{62}{16}
\][/tex]
Simplifying the left side:
[tex]\[
b^{7 - 3.5} = b^{3.5} = \frac{62}{16}
\][/tex]
Calculating [tex]\( \frac{62}{16} \)[/tex]:
[tex]\[
b^{3.5} = 3.875
\][/tex]
3. Solve for [tex]\( b \)[/tex]:
To find [tex]\( b \)[/tex], take the 3.5th root of 3.875:
[tex]\[
b = 3.875^{\frac{1}{3.5}} \approx 1.472
\][/tex]
4. Solve for [tex]\( a \)[/tex] using one of the original equations:
Using [tex]\( a \cdot b^{3.5} = 16 \)[/tex]:
[tex]\[
a = \frac{16}{b^{3.5}}
\][/tex]
Substituting the approximate value of [tex]\( b \)[/tex]:
[tex]\[
a = \frac{16}{(1.472)^{3.5}} \approx 4.129
\][/tex]
5. Calculate [tex]\( f(10.5) \)[/tex]:
Finally, use the function with the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(10.5) = a \cdot b^{10.5}
\][/tex]
Substituting the values:
[tex]\[
f(10.5) = 4.129 \times (1.472)^{10.5} \approx 240.25
\][/tex]
Thus, the value of [tex]\( f(10.5) \)[/tex] to the nearest hundredth is 240.25.
1. Set up the equations using the given points:
- For [tex]\( f(3.5) = 16 \)[/tex]:
[tex]\[
a \cdot b^{3.5} = 16
\][/tex]
- For [tex]\( f(7) = 62 \)[/tex]:
[tex]\[
a \cdot b^7 = 62
\][/tex]
2. Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^7}{a \cdot b^{3.5}} = \frac{62}{16}
\][/tex]
Simplifying the left side:
[tex]\[
b^{7 - 3.5} = b^{3.5} = \frac{62}{16}
\][/tex]
Calculating [tex]\( \frac{62}{16} \)[/tex]:
[tex]\[
b^{3.5} = 3.875
\][/tex]
3. Solve for [tex]\( b \)[/tex]:
To find [tex]\( b \)[/tex], take the 3.5th root of 3.875:
[tex]\[
b = 3.875^{\frac{1}{3.5}} \approx 1.472
\][/tex]
4. Solve for [tex]\( a \)[/tex] using one of the original equations:
Using [tex]\( a \cdot b^{3.5} = 16 \)[/tex]:
[tex]\[
a = \frac{16}{b^{3.5}}
\][/tex]
Substituting the approximate value of [tex]\( b \)[/tex]:
[tex]\[
a = \frac{16}{(1.472)^{3.5}} \approx 4.129
\][/tex]
5. Calculate [tex]\( f(10.5) \)[/tex]:
Finally, use the function with the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(10.5) = a \cdot b^{10.5}
\][/tex]
Substituting the values:
[tex]\[
f(10.5) = 4.129 \times (1.472)^{10.5} \approx 240.25
\][/tex]
Thus, the value of [tex]\( f(10.5) \)[/tex] to the nearest hundredth is 240.25.
