Question

If f(x) is an exponential function where f(1.5) = 7 and f(8.5) = 46, then find the value of f(14), to the nearest hundredth.

Answer 1

Answer:

$f(14)=201.932$

Step-by-step explanation:

Set up a system of equations and solve for "b"

$\left \{ {{46=ab^{8.5}} \atop {7=ab^{1.5}}} \right\\ \\\left \{ {{\frac{46}{b^{8.5}}=a } \atop {\frac{7}{b^{1.5}}=a} \right\\\\\frac{46}{b^{8.5}}=\frac{7}{b^{1.5}}\\ \\46b^{1.5}=7b^{8.5}\\\\46=7b^7\\\\\frac{46}{7}=b^7\\ \\b=1.308604899$

Determine the value of "a" using "b"

$y=ab^x\\\\46=a(1.308604899)^{8.5}\\\\46=9.837224985a\\\\a=4.676115477$

Find f(14) using the new function

$f(x)=4.676115477(1.308604899)^x\\\\f(14)=4.676115477(1.308604899)^{14}\\\\f(14)\approx201.932$