Answer:
[tex]y=106.656*(0.816)^x[/tex]
Step-by-step explanation:
The exponential equation will be of the form:
[tex]y=ba^x.[/tex]
Now from the information give we have two points:
at [tex]x=3,[/tex] [tex]y=58.[/tex]
at [tex]x=10,[/tex] [tex]y=14.[/tex]
Thus we have two equations
[tex](1).58=ba^3,[/tex]
[tex](2).14=ba^{10}.[/tex]
From equation [tex](1)[/tex] we solve for [tex]b[/tex]:
[tex]b=\frac{58}{a^3},[/tex]
and put this value into equation [tex](2)[/tex] and we get:
[tex]14=\frac{58}{a^3}*a^{10}=58a^7,[/tex]
and solve for [tex]a:[/tex]
[tex]a=\sqrt[7]{\frac{14}{58} }=0.8162.[/tex]
[tex]\boxed{a=0.816.}[/tex]
We now put this value into equation (1) and solve for [tex]b[/tex]:
[tex]\boxed{b=\frac{58}{(0.8162)^3} =106.656.}[/tex].
With values of [tex]a[/tex] and [tex]b[/tex] in hand, we have our exponential function:
[tex]\boxed{y=106.656*(0.816)^x.}[/tex]
