Answer :
Using the given points and solving the exponential function, f(10.5) is approximately 363.37 to the nearest hundredth.
To find the value of f(10.5) for the exponential function f(x), we need to determine the equation of the function first. Exponential functions are typically represented in the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
Where:
- a is the initial value or the value of the function when x = 0.
- b is the base of the exponential function
We can use the given points (3, 11) and (5.5, 57) to form a system of equations and solve for a and b.
From the point (3, 11):
[tex]\[ 11 = a \cdot b^3 \][/tex]
From the point (5.5, 57):
[tex]\[ 57 = a \cdot b^{5.5} \][/tex]
Now, we can solve this system of equations to find the values of a and b. Once we have a and b, we can find f(10.5).
Let's do the calculations.
We have the equations:
[tex]\[ 11 = a \cdot b^3 \][/tex]
[tex]\[ 57 = a \cdot b^{5.5} \][/tex]
To eliminate a, divide the second equation by the first:
[tex]\[ \frac{57}{11} = \frac{a \cdot b^{5.5}}{a \cdot b^3} \][/tex]
[tex]\[ \frac{57}{11} = b^{5.5 - 3} \][/tex]
[tex]\[ \frac{57}{11} = b^{2.5} \][/tex]
Now, take the square root of both sides to solve for b:
[tex]\[ b = \sqrt[2.5]{\frac{57}{11}} \][/tex]
[tex]\[ b \approx 2.37 \][/tex]
Now that we have found b, we can substitute it into one of the original equations to find a. Let's use the first equation:
[tex]\[ 11 = a \cdot (2.37)^3 \][/tex]
[tex]\[ 11 = a \cdot 13.49 \][/tex]
[tex]\[ a \approx \frac{11}{13.49} \][/tex]
[tex]\[ a \approx 0.815 \][/tex]
Now that we have found a and b, we can find f(10.5):
[tex]\[ f(10.5) = 0.815 \cdot (2.37)^{10.5} \][/tex]
[tex]\[ f(10.5) \approx 0.815 \cdot 446.07 \][/tex]
[tex]\[ f(10.5) \approx 363.37 \][/tex]
So, f(10.5) is approximately 363.37 to the nearest hundredth.
