If f, of, xf(x) is an exponential function where f, of, 4, equals, 17f(4)=17 and f, of, 12, point, 5, equals, 31f(12.5)=31, then find the value of f, of, 18, point, 5f(18.5), to the nearest hundredth.
To find the value of f(18.5), we need to find the equation that represents the exponential function f(x).
We know that f(4) = 17 and f(12.5) = 31. Using these two data points, we can set up a system of equations to solve for the values of a and b in the equation f(x) = a * b^x:
17 = a * b^4 (equation 1)
31 = a * b^12.5 (equation 2)
To solve this system of equations, we can divide equation 2 by equation 1:
31 / 17 = (a * b^12.5) / (a * b^4)
1.8235 = b^(12.5-4)
1.8235 = b^8.5
Taking the logarithm of both sides (to base b) to solve for b:
log_b(1.8235) = 8.5
Since b^8.5 = 1.8235, b ≈ 1.1846.
Substituting this value of b into equation 1, we can solve for a:
17 = a * (1.1846)^4
17 = a * 1.9543
a ≈ 8.7047
Therefore, the equation of the exponential function f(x) is approximately f(x) ≈ 8.7047 * (1.1846)^x.
Now we can find the value of f(18.5):
f(18.5) ≈ 8.7047 * (1.1846)^(18.5)
≈ 8.7047 * 17.8236
≈ 155.16
Therefore, the value of f(18.5) to the nearest hundredth is approximately 155.16.