Question
If f(x) is an exponential function of the form of y=ab^{x} where f(4)=16 and f(4.5)=54, then find the value of f(5), to the nearest hundredth.
Ask by Valdez Powers.
Jan 23,2025 17:37
UpStudy AI Solution
Tutor-Verified Answer
Answer
f(5) = 182.25
Solution
Given that \( f(x) \) is an exponential function of the form \( y = ab^x \), where \( f(4) = 16 \) and \( f(4.5) = 54 \), we need to find the value of \( f(5) \) to the nearest hundredth.
To find the value of \( f(5) \), we first need to determine the values of \( a \) and \( b \) using the given information.
From the given information, we have:
1. \( f(4) = 16 \)
2. \( f(4.5) = 54 \)
We can use these two equations to solve for \( a \) and \( b \).
Let's start by using the first equation \( f(4) = 16 \) to find the value of \( a \):
\[ 16 = ab^4 \]
Now, let's use the second equation \( f(4.5) = 54 \) to find the value of \( b \):
\[ 54 = ab^{4.5} \]
We can solve these two equations simultaneously to find the values of \( a \) and \( b \).
Let's solve these equations to find the values of \( a \) and \( b \) and then use them to find the value of \( f(5) \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}16=ab^{4}\\54=ab^{4.5}\end{array}\right.\)
- step1: Calculate:
\(\left\{ \begin{array}{l}16=ab^{4}\\54=ab^{\frac{9}{2}}\end{array}\right.\)
- step2: Solve the equation:
\(\left\{ \begin{array}{l}a=\frac{16}{b^{4}}\\54=ab^{\frac{9}{2}}\end{array}\right.\)
- step3: Substitute the value of \(a:\)
\(54=\frac{16}{b^{4}}\times b^{\frac{9}{2}}\)
- step4: Reduce the fraction:
\(54=16b^{\frac{1}{2}}\)
- step5: Swap the sides:
\(16b^{\frac{1}{2}}=54\)
- step6: Divide both sides:
\(\frac{16b^{\frac{1}{2}}}{16}=\frac{54}{16}\)
- step7: Divide the numbers:
\(b^{\frac{1}{2}}=\frac{27}{8}\)
- step8: Raise both sides to the \(2\)-th power\(:\)
\(\left(b^{\frac{1}{2}}\right)^{2}=\left(\frac{27}{8}\right)^{2}\)
- step9: Evaluate the power:
\(b=\frac{729}{64}\)
- step10: Substitute the value of \(b:\)
\(a=\frac{16}{\left(\frac{729}{64}\right)^{4}}\)
- step11: Simplify:
\(a=\frac{2^{28}}{729^{4}}\)
- step12: Calculate:
\(\left\{ \begin{array}{l}a=\frac{2^{28}}{729^{4}}\\b=\frac{729}{64}\end{array}\right.\)
- step13: Check the solution:
\(\left\{ \begin{array}{l}a=\frac{2^{28}}{729^{4}}\\b=\frac{729}{64}\end{array}\right.\)
- step14: Rewrite:
\(\left(a,b\right) = \left(\frac{2^{28}}{729^{4}},\frac{729}{64}\right)\)
The solution to the system of equations is \( a = \frac{2^{28}}{729^{4}} \) and \( b = \frac{729}{64} \).
Now that we have the values of \( a \) and \( b \), we can find the value of \( f(5) \) using the exponential function \( f(x) = ab^x \).
Substitute the values of \( a \) and \( b \) into the function to find \( f(5) \):
\[ f(5) = \frac{2^{28}}{729^{4}} \times \left(\frac{729}{64}\right)^5 \]
Let's calculate the value of \( f(5) \) to the nearest hundredth.
Calculate the value by following steps:
- step0: Calculate:
\(\frac{2^{28}}{729^{4}}\times \left(\frac{729}{64}\right)^{5}\)
- step1: Evaluate the power:
\(\frac{2^{28}}{729^{4}}\times \frac{729^{5}}{64^{5}}\)
- step2: Rewrite the expression:
\(\frac{2^{28}}{729^{4}}\times \frac{729^{5}}{2^{30}}\)
- step3: Reduce the numbers:
\(1\times \frac{729}{2^{2}}\)
- step4: Multiply:
\(\frac{729}{2^{2}}\)
- step5: Evaluate the power:
\(\frac{729}{4}\)
The value of \( f(5) \) to the nearest hundredth is 182.25.
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Explain 
Simplify this solution Beyond the Answer
To solve for \( f(5) \), we start with the given form of the function \( f(x) = a b^{x} \) and the provided values \( f(4) = 16 \) and \( f(4.5) = 54 \).
Setting up the equations with the known values:
1. From \( f(4) = 16 \):
\[
a b^4 = 16 \quad (1)
\]
2. From \( f(4.5) = 54 \):
\[
a b^{4.5} = 54 \quad (2)
\]
Next, we can express \( b^{4.5} \) in terms of \( b^4 \):
\[
b^{4.5} = b^4 \cdot b^{0.5}
\]
Substituting this into equation (2):
\[
a b^4 \cdot b^{0.5} = 54
\]
Now, replacing \( a b^4 \) from equation (1):
\[
16 \cdot b^{0.5} = 54
\]
Solving for \( b^{0.5} \):
\[
b^{0.5} = \frac{54}{16} = 3.375
\]
To find \( b \), we square both sides:
\[
b = (3.375)^2 = 11.390625
\]
Now, substitute \( b \) back into equation (1) to find \( a \):
\[
a (11.390625)^4 = 16
\]
Calculating \( (11.390625)^4 \):
\[
(11.390625)^4 \approx 16484.96449
\]
Then,
\[
a \cdot 16484.96449 = 16 \implies a = \frac{16}{16484.96449} \approx 0.000969
\]
Now we can express \( f(x) \) fully:
\[
f(x) = 0.000969 \cdot (11.390625)^x
\]
Finally, we need to find \( f(5) \):
\[
f(5) = 0.000969 \cdot (11.390625)^5
\]
Calculating \( (11.390625)^5 \):
\[
(11.390625)^5 \approx 187384.6237
\]
Then,
\[
f(5) \approx 0.000969 \cdot 187384.6237 \approx 181.539
\]
Thus, the value of \( f(5) \) to the nearest hundredth is:
\[
\boxed{181.54}
\]
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