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Answer :
To find the value of [tex]\( f(7) \)[/tex] for the exponential function [tex]\( y = a \cdot b^x \)[/tex], we are given two points: [tex]\( f(4.5) = 17 \)[/tex] and [tex]\( f(5) = 94 \)[/tex].
1. Set up the equations for the given points:
- From [tex]\( f(4.5) = 17 \)[/tex], we have:
[tex]\[ 17 = a \cdot b^{4.5} \][/tex]
- From [tex]\( f(5) = 94 \)[/tex], we have:
[tex]\[ 94 = a \cdot b^5 \][/tex]
2. Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{94}{17} = \frac{a \cdot b^5}{a \cdot b^{4.5}}
\][/tex]
Simplifying gives us:
[tex]\[
\frac{94}{17} = b^{5 - 4.5} = b^{0.5}
\][/tex]
3. Solve for [tex]\( b \)[/tex]:
[tex]\[
b = \left( \frac{94}{17} \right)^2
\][/tex]
Calculating the above gives us:
[tex]\[
b \approx 30.57
\][/tex]
4. Use the value of [tex]\( b \)[/tex] to solve for [tex]\( a \)[/tex] using one of the original equations, for example, [tex]\( f(4.5) = 17 \)[/tex]:
[tex]\[
17 = a \cdot (30.57)^{4.5}
\][/tex]
[tex]\[
a = \frac{17}{(30.57)^{4.5}}
\][/tex]
Calculating gives us:
[tex]\[
a \approx 3.52 \times 10^{-6}
\][/tex]
5. Use the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to find [tex]\( f(7) \)[/tex]:
[tex]\[
f(7) = a \cdot b^7
\][/tex]
Substitute the values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(7) = 3.52 \times 10^{-6} \cdot (30.57)^7
\][/tex]
Calculating this gives:
[tex]\[
f(7) \approx 87870.60
\][/tex]
Thus, the value of [tex]\( f(7) \)[/tex] is approximately [tex]\( 87870.60 \)[/tex] when rounded to the nearest hundredth.
1. Set up the equations for the given points:
- From [tex]\( f(4.5) = 17 \)[/tex], we have:
[tex]\[ 17 = a \cdot b^{4.5} \][/tex]
- From [tex]\( f(5) = 94 \)[/tex], we have:
[tex]\[ 94 = a \cdot b^5 \][/tex]
2. Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{94}{17} = \frac{a \cdot b^5}{a \cdot b^{4.5}}
\][/tex]
Simplifying gives us:
[tex]\[
\frac{94}{17} = b^{5 - 4.5} = b^{0.5}
\][/tex]
3. Solve for [tex]\( b \)[/tex]:
[tex]\[
b = \left( \frac{94}{17} \right)^2
\][/tex]
Calculating the above gives us:
[tex]\[
b \approx 30.57
\][/tex]
4. Use the value of [tex]\( b \)[/tex] to solve for [tex]\( a \)[/tex] using one of the original equations, for example, [tex]\( f(4.5) = 17 \)[/tex]:
[tex]\[
17 = a \cdot (30.57)^{4.5}
\][/tex]
[tex]\[
a = \frac{17}{(30.57)^{4.5}}
\][/tex]
Calculating gives us:
[tex]\[
a \approx 3.52 \times 10^{-6}
\][/tex]
5. Use the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to find [tex]\( f(7) \)[/tex]:
[tex]\[
f(7) = a \cdot b^7
\][/tex]
Substitute the values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(7) = 3.52 \times 10^{-6} \cdot (30.57)^7
\][/tex]
Calculating this gives:
[tex]\[
f(7) \approx 87870.60
\][/tex]
Thus, the value of [tex]\( f(7) \)[/tex] is approximately [tex]\( 87870.60 \)[/tex] when rounded to the nearest hundredth.
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