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Answer :
Let's solve this problem step-by-step to find the value of [tex]\( f(7) \)[/tex].
We know that [tex]\( f(x) = a \cdot b^x \)[/tex] and we're given two points: [tex]\( f(4.5) = 17 \)[/tex] and [tex]\( f(5) = 94 \)[/tex].
Step 1: Set up the equations using the given points.
From the equation at [tex]\( x = 4.5 \)[/tex], we have:
[tex]\[ a \cdot b^{4.5} = 17 \][/tex]
From the equation at [tex]\( x = 5 \)[/tex], we have:
[tex]\[ a \cdot b^5 = 94 \][/tex]
Step 2: Solve for [tex]\( b \)[/tex].
Divide the second equation by the first equation to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^5}{a \cdot b^{4.5}} = \frac{94}{17}
\][/tex]
This simplifies to:
[tex]\[
b^{0.5} = \frac{94}{17}
\][/tex]
Calculate [tex]\( b \)[/tex] by squaring both sides:
[tex]\[
b = \left(\frac{94}{17}\right)^2
\][/tex]
Now compute this value:
[tex]\[ \frac{94}{17} \approx 5.5294 \][/tex]
Square this value:
[tex]\[ b = 5.5294^2 \approx 30.5778 \][/tex]
Step 3: Solve for [tex]\( a \)[/tex] using one of the original equations.
Substitute the value of [tex]\( b \)[/tex] back into one of the original equations, say [tex]\( a \cdot b^{4.5} = 17 \)[/tex]:
[tex]\[
a \cdot (30.5778)^{4.5} = 17
\][/tex]
Now solve for [tex]\( a \)[/tex]:
First, calculate [tex]\( (30.5778)^{4.5} \)[/tex]:
[tex]\[
(30.5778)^{4.5} \approx 50639.38
\][/tex]
Now solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{17}{50639.38} \approx 0.0003356
\][/tex]
Step 4: Calculate [tex]\( f(7) \)[/tex].
Finally, use the function [tex]\( f(x) = a \cdot b^x \)[/tex] to find [tex]\( f(7) \)[/tex]:
[tex]\[
f(7) = 0.0003356 \cdot (30.5778)^7
\][/tex]
Calculate [tex]\( (30.5778)^7 \)[/tex]:
[tex]\[ (30.5778)^7 \approx 2.2914 \times 10^{7} \][/tex]
Now, compute [tex]\( f(7) \)[/tex]:
[tex]\[
f(7) \approx 0.0003356 \times 2.2914 \times 10^{7} \approx 7692.94
\][/tex]
Thus, the value of [tex]\( f(7) \)[/tex], to the nearest hundredth, is:
[tex]\[ \boxed{7692.94} \][/tex]
We know that [tex]\( f(x) = a \cdot b^x \)[/tex] and we're given two points: [tex]\( f(4.5) = 17 \)[/tex] and [tex]\( f(5) = 94 \)[/tex].
Step 1: Set up the equations using the given points.
From the equation at [tex]\( x = 4.5 \)[/tex], we have:
[tex]\[ a \cdot b^{4.5} = 17 \][/tex]
From the equation at [tex]\( x = 5 \)[/tex], we have:
[tex]\[ a \cdot b^5 = 94 \][/tex]
Step 2: Solve for [tex]\( b \)[/tex].
Divide the second equation by the first equation to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^5}{a \cdot b^{4.5}} = \frac{94}{17}
\][/tex]
This simplifies to:
[tex]\[
b^{0.5} = \frac{94}{17}
\][/tex]
Calculate [tex]\( b \)[/tex] by squaring both sides:
[tex]\[
b = \left(\frac{94}{17}\right)^2
\][/tex]
Now compute this value:
[tex]\[ \frac{94}{17} \approx 5.5294 \][/tex]
Square this value:
[tex]\[ b = 5.5294^2 \approx 30.5778 \][/tex]
Step 3: Solve for [tex]\( a \)[/tex] using one of the original equations.
Substitute the value of [tex]\( b \)[/tex] back into one of the original equations, say [tex]\( a \cdot b^{4.5} = 17 \)[/tex]:
[tex]\[
a \cdot (30.5778)^{4.5} = 17
\][/tex]
Now solve for [tex]\( a \)[/tex]:
First, calculate [tex]\( (30.5778)^{4.5} \)[/tex]:
[tex]\[
(30.5778)^{4.5} \approx 50639.38
\][/tex]
Now solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{17}{50639.38} \approx 0.0003356
\][/tex]
Step 4: Calculate [tex]\( f(7) \)[/tex].
Finally, use the function [tex]\( f(x) = a \cdot b^x \)[/tex] to find [tex]\( f(7) \)[/tex]:
[tex]\[
f(7) = 0.0003356 \cdot (30.5778)^7
\][/tex]
Calculate [tex]\( (30.5778)^7 \)[/tex]:
[tex]\[ (30.5778)^7 \approx 2.2914 \times 10^{7} \][/tex]
Now, compute [tex]\( f(7) \)[/tex]:
[tex]\[
f(7) \approx 0.0003356 \times 2.2914 \times 10^{7} \approx 7692.94
\][/tex]
Thus, the value of [tex]\( f(7) \)[/tex], to the nearest hundredth, is:
[tex]\[ \boxed{7692.94} \][/tex]
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