Answer:
[tex]f(10) = 173[/tex]
Step-by-step explanation:
Given
Exponential Function
[tex](x_1,y_1) = (3,10.125)[/tex]
[tex](x_2,y_2) = (6,34.2)[/tex]
Required
Determine f(10)
We have that
[tex]y = ab^x[/tex]
First, we need to solve for the values of a and b
For [tex](x_1,y_1) = (3,10.125)[/tex]
[tex]10.125 = ab^3[/tex] --- (1)
For [tex](x_2,y_2) = (6,34.2)[/tex]
[tex]34.2 = ab^6[/tex] ---- (2)
Divide (2) by (1)
[tex]\frac{34.2}{10.125} = \frac{ab^6}{ab^3}[/tex]
[tex]\frac{34.2}{10.125} = \frac{b^6}{b^3}[/tex]
[tex]3.38= b^{6-3}[/tex]
[tex]3.38= b^{3}[/tex]
Take cube root of both sides
[tex]b = \sqrt[3]{3.38}[/tex]
[tex]b = 1.5[/tex]
Substitute 1.5 for b in [tex]10.125 = ab^3[/tex]
[tex]10.125 = a * 1.5^3[/tex]
[tex]10.125 = a * 3.375[/tex]
Solve for a
[tex]a = \frac{10.125}{3.375}[/tex]
[tex]a = 3[/tex]
To solve for f(10).
This implies that x = 10
So, we have:
[tex]y = ab^x[/tex] which becomes
[tex]y = 3 * 1.5^{10[/tex]
[tex]y = 3 * 57.6650390625[/tex]
[tex]y = 172.995117188[/tex]
[tex]y = 173[/tex] -- approximated
Hence:
[tex]f(10) = 173[/tex]
