SOLUTION: Given that y=ax^n - 23 and that y=4 when x=3 and y=220 when x=9 find the value of a and n.

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Given that y=ax^n - 23 and that y=4 when x=3 and y=220 when x=9 find the value of a and n.      Log On


   

Question 818780: Given that y=ax^n - 23 and that y=4 when x=3 and y=220 when x=9 find the value of a and n.
Answer by LinnW(1048) About Me  (Show Source):
You can put this solution on YOUR website!
Given that y=ax^n - 23 and that y=4 when x=3 and y=220 when x=9 find the value of a and n.
y=ax^n - 23 and that y=4 when x=3 and y=220 when x=9 find the value of a and n.
4 = a(3)^n
and
220 = a(9)^n
Let's solve each equation for a
4 = a(3)^n
divide each side by (3)^n an we have
a=4%2F%28%283%29%5En%29
Similarly 220 = a(9)^n can be written
a=220%2F%28%289%29%5En%29
Setting the right hand sides to be equal,
4%2F%28%283%29%5En%29=220%2F%28%289%29%5En%29
Taking the cross products we have
4%2A%28%289%29%5En%29=220%2A%28%283%29%5En%29
Divide each side by 4
%28%289%29%5En%29=%28220%2F4%29%2A%28%283%29%5En%29or%28%289%29%5En%29=55%2A%28%283%29%5En%29
Divide each side by %28%283%29%5En%29
= %28%289%29%5En%29%2F%28%283%29%5En%29=55or %28%283%5E2%29%5En%29%2F%28%283%29%5En%29=55
= %28%283%5En%29%5E2%29%2F%28%283%29%5En%29=55So,%283%29%5En=55
Take the log of each side
log%28%283%29%5En%29=log%2855%29so, n%2Alog%28%283%29%29=log%2855%29
Divide each side by log(3)
n=%28log%2855%29%29%2F%28log%283%29%29=3.647632
Now substitute in a=4%2F%28%283%29%5En%29,a=4%2F%28%283%29%5E3.647632%29,a=4%2F%2855%29
a=0.072727
Let's check the second equation, 220 = a(9)^n
220 ?= (0.072727)(9^3.647632)
220 ?= (0.072727)(3025.001)
= 219.9992 So considering rounding error we seem to be correct.