I am asked to find an exponential equation that passes through $(2, 2.25)$ and $(5, 60.75)$. My textbook says the solution it's $y=0.25(3)^x$ whereas I got $y=0.028(9)^x$. Here is my working:
Express each coordinate in terms of the exponential function:
$$2.25=ab^2$$
$$60.75=ab^5$$
Solve for $a$ in terms of $b$ in the first equation:
$$a=\frac{2.25}{b^2}$$
$$a=2.25b^{-2}$$
Substitute $a$ in the second equation and solve for $b$:
$$60.75=2.25b^{-2}b^5$$
$$60.75=2.25b^3$$
$$b^3=\frac{60.75}{2.25}$$
$$b=27^{\frac{1}{3}}$$
$$b=9$$
Use the value of b in the first equation to solve for the value of a:
$$a=2.25b^{-2}$$
$$a=2.25(9)^{-2}$$
$$a=\frac{2.25}{9^2}$$
$$a=0.028$$
Thus my solution is:
$$y=0.028(9)^x$$
Where did I go wrong and how can I arrive at $y=0.25(3)^x$?
Best Answer
You have mistakenly wrote: $$b = 27^{\frac{1}{3}}$$ $$ b = 9 $$
When in fact $b = 27^{\frac{1}{3}} = 3$.