Answer:
The value of [tex]f(6.5)[/tex] is 28.22.
Step-by-step explanation:
The exponential function is modelled after this model:
[tex]y = A\cdot e^{B\cdot x}[/tex] (1)
Where:
[tex]A, B[/tex] - Coefficients.
[tex]x[/tex] - Independent variable.
[tex]y[/tex] - Dependent variable.
By using all information from statement, we have the following system of equations: [tex](x_{1},y_{1}) = (4.5, 24)[/tex], [tex](x_{2}, y_{2}) = (14.5, 54)[/tex]
[tex]A\cdot e^{4.5\cdot B} = 24[/tex] (2)
[tex]A\cdot e^{14.5\cdot B} = 54[/tex] (3)
By dividing (3) by (2), we calculate [tex]B[/tex]:
[tex]e^{10\cdot B} = \frac{9}{4}[/tex]
[tex]10\cdot B = \ln \frac{9}{4}[/tex]
[tex]B = \frac{1}{10}\cdot \ln \frac{9}{4}[/tex]
[tex]B \approx 0.081[/tex]
If we know that [tex](x,y) = (4.5, 24)[/tex] and [tex]B \approx 0.081[/tex], then the value of [tex]A[/tex] is:
[tex]A = \frac{y}{e^{B\cdot x }}[/tex]
[tex]A \approx 16.669[/tex]
If we know that [tex]x = 6.5[/tex], then the value of [tex]y[/tex] is:
[tex]y = 16.669\cdot e^{0.081\cdot x}[/tex] (4)
[tex]y \approx 28.221[/tex]
The value of [tex]f(6.5)[/tex] is 28.22.
