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Answer :
Let's solve the problem step by step.
The exponential function is given by [tex]\( f(x) = a \cdot b^x \)[/tex]. We have the values:
- [tex]\( f(0.5) = 26 \)[/tex]
- [tex]\( f(1) = 66 \)[/tex]
Step 1: Find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
Since [tex]\( f(1) = 66 \)[/tex], we have:
[tex]\[ a \cdot b^1 = 66 \][/tex]
[tex]\[ a \cdot b = 66 \][/tex]
Since [tex]\( f(0.5) = 26 \)[/tex], we have:
[tex]\[ a \cdot b^{0.5} = 26 \][/tex]
Now, let's solve these two equations. First, let's express [tex]\( a \)[/tex] from the second equation:
[tex]\[ a = \frac{26}{b^{0.5}} \][/tex]
Substitute this expression for [tex]\( a \)[/tex] in the first equation:
[tex]\[ \frac{26}{b^{0.5}} \cdot b = 66 \][/tex]
This simplifies to:
[tex]\[ 26 \cdot b^{0.5} = 66 \][/tex]
To solve for [tex]\( b^{0.5} \)[/tex]:
[tex]\[ b^{0.5} = \frac{66}{26} \][/tex]
[tex]\[ b^{0.5} = \frac{33}{13} \][/tex]
Now square both sides to find [tex]\( b \)[/tex]:
[tex]\[ b = \left(\frac{33}{13}\right)^2 \][/tex]
Step 2: Calculate the value of [tex]\( b \)[/tex].
[tex]\[ b = \left(\frac{33}{13}\right)^2 \approx 6.448717948717949 \][/tex]
Step 3: Find the value of [tex]\( a \)[/tex].
Using [tex]\( a \cdot b = 66 \)[/tex]:
[tex]\[ a = \frac{66}{b} \][/tex]
Substitute the value of [tex]\( b \)[/tex]:
[tex]\[ a = \frac{66}{6.448717948717949} \approx 10.23 \][/tex]
Step 4: Find [tex]\( f(1.5) \)[/tex].
Finally, we use:
[tex]\[ f(1.5) = a \cdot b^{1.5} \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ f(1.5) \approx 10.23 \cdot (6.448717948717949)^{1.5} \][/tex]
Now calculate this value:
[tex]\[ f(1.5) \approx 10.23 \cdot 16.44802257 \approx 168.35 \][/tex]
So, the value of [tex]\( f(1.5) \)[/tex], rounded to the nearest hundredth, is [tex]\( 168.35 \)[/tex].
The exponential function is given by [tex]\( f(x) = a \cdot b^x \)[/tex]. We have the values:
- [tex]\( f(0.5) = 26 \)[/tex]
- [tex]\( f(1) = 66 \)[/tex]
Step 1: Find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
Since [tex]\( f(1) = 66 \)[/tex], we have:
[tex]\[ a \cdot b^1 = 66 \][/tex]
[tex]\[ a \cdot b = 66 \][/tex]
Since [tex]\( f(0.5) = 26 \)[/tex], we have:
[tex]\[ a \cdot b^{0.5} = 26 \][/tex]
Now, let's solve these two equations. First, let's express [tex]\( a \)[/tex] from the second equation:
[tex]\[ a = \frac{26}{b^{0.5}} \][/tex]
Substitute this expression for [tex]\( a \)[/tex] in the first equation:
[tex]\[ \frac{26}{b^{0.5}} \cdot b = 66 \][/tex]
This simplifies to:
[tex]\[ 26 \cdot b^{0.5} = 66 \][/tex]
To solve for [tex]\( b^{0.5} \)[/tex]:
[tex]\[ b^{0.5} = \frac{66}{26} \][/tex]
[tex]\[ b^{0.5} = \frac{33}{13} \][/tex]
Now square both sides to find [tex]\( b \)[/tex]:
[tex]\[ b = \left(\frac{33}{13}\right)^2 \][/tex]
Step 2: Calculate the value of [tex]\( b \)[/tex].
[tex]\[ b = \left(\frac{33}{13}\right)^2 \approx 6.448717948717949 \][/tex]
Step 3: Find the value of [tex]\( a \)[/tex].
Using [tex]\( a \cdot b = 66 \)[/tex]:
[tex]\[ a = \frac{66}{b} \][/tex]
Substitute the value of [tex]\( b \)[/tex]:
[tex]\[ a = \frac{66}{6.448717948717949} \approx 10.23 \][/tex]
Step 4: Find [tex]\( f(1.5) \)[/tex].
Finally, we use:
[tex]\[ f(1.5) = a \cdot b^{1.5} \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ f(1.5) \approx 10.23 \cdot (6.448717948717949)^{1.5} \][/tex]
Now calculate this value:
[tex]\[ f(1.5) \approx 10.23 \cdot 16.44802257 \approx 168.35 \][/tex]
So, the value of [tex]\( f(1.5) \)[/tex], rounded to the nearest hundredth, is [tex]\( 168.35 \)[/tex].
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