Question

If f(x) is an exponential function where f(4.5) = 24 and f(14.5) = 54, then find

Answer 1

Answer:

The value of $f(6.5)$ is 28.22.

Step-by-step explanation:

The exponential function is modelled after this model:

$y = A\cdot e^{B\cdot x}$ (1)

Where:

$A, B$ - Coefficients.

$x$ - Independent variable.

$y$ - Dependent variable.

By using all information from statement, we have the following system of equations: $(x_{1},y_{1}) = (4.5, 24)$, $(x_{2}, y_{2}) = (14.5, 54)$

$A\cdot e^{4.5\cdot B} = 24$ (2)

$A\cdot e^{14.5\cdot B} = 54$ (3)

By dividing (3) by (2), we calculate $B$:

$e^{10\cdot B} = \frac{9}{4}$

$10\cdot B = \ln \frac{9}{4}$

$B = \frac{1}{10}\cdot \ln \frac{9}{4}$

$B \approx 0.081$

If we know that $(x,y) = (4.5, 24)$ and $B \approx 0.081$, then the value of $A$ is:

$A = \frac{y}{e^{B\cdot x }}$

$A \approx 16.669$

If we know that $x = 6.5$, then the value of $y$ is:

$y = 16.669\cdot e^{0.081\cdot x}$ (4)

$y \approx 28.221$

The value of $f(6.5)$ is 28.22.