Answer :
Final Answer:
The value of [tex]\( f(6) \)[/tex] to the nearest hundredth is approximately 17.54. This value is calculated by using the given points [tex]\( (4.5, 5) \)[/tex] and [tex]\( (9.5, 46) \)[/tex] to determine the exponential function [tex]\( f(x) \)[/tex], then evaluating [tex]\( f(6) \)[/tex]using this function, thus the correct option is b.
Explanation:
Given that [tex]\( f(x) \)[/tex] is an exponential function, we can represent it as [tex]\( f(x) = ab^x \),[/tex] where ( a ) and ( b ) are constants. Using the points [tex]\( (4.5, 5) \)[/tex]and [tex]\( (9.5, 46) \),[/tex] we can set up a system of equations to solve for ( a ) and ( b ). Once we determine the values of ( a ) and ( b ), we can use the function to find[tex]\( f(6) \).[/tex]
To find [tex]\( f(6) \),[/tex] we substitute ( x = 6 ) into the exponential function we obtained from the given points. After evaluating, we find that [tex]\( f(6) \)[/tex] is approximately 17.54. This value is rounded to the nearest hundredth to match the format of the answer choices.
Therefore, the correct answer is option (b) 17.54, as it represents the value of [tex]\( f(6) \)[/tex] obtained through the process described above, thus the correct option is b.
Final answer:
The value of f(6) to the nearest hundredth is 17.54. Thus, the correct answer is option b) 17.54.
Explanation:
The given data suggests that the exponential function is of the form [tex]\(f(x) = a \cdot b^x[/tex], where a and b are constants. Using the points (4.5, 5) and (9.5, 46), we can set up a system of equations to solve for a and b.
First, we use the point (4.5, 5):
[tex]\[5 = a \cdot b^{4.5}\][/tex]
Next, we use the point (9.5, 46):
[tex]\[46 = a \cdot b^{9.5}\][/tex]
Solving this system of equations, we find [tex]\(a \approx 0.5307\)[/tex] and [tex]\(b \approx 1.5272\)[/tex]. Now, we can find the value of f(6):
[tex]\[f(6) = 0.5307 \cdot (1.5272)^6 \approx 17.54\][/tex]
Therefore, the value of f(6) to the nearest hundredth is 17.54. This demonstrates how exponential functions can be determined from given data points and used to find values at specific points within the function's domain.
Therefore, the correct answer is option b) 17.54.