2009 AMC 12B Problem 15

Question

Assume

0 \lt r \lt 3.

Below are five equations for

x.

Which equation has the largest solution

x?

3(1 + r)^x = 7

3(1 + \frac{r}{10})^x = 7

3(1 + 2r)^x = 7

3(1 + \sqrt{r})^x = 7

3(1 + \frac{1}{r})^x = 7

Solving each gives

x = \dfrac{\log(\frac{7}{3})}{\log(1 + f(r))}

; a smaller base makes

x

larger

Compare the values

r,\ \frac{r}{10},\ 2r,\ \sqrt{r},\ \frac{1}{r}

for

0 \lt r \lt 3

and find the smallest

Each equation gives

x = \dfrac{\log(\frac{7}{3})}{\log(1 + f(r))},

which is largest when the positive quantity

f(r)

is smallest.

For

0 \lt r \lt 3,

we have

\dfrac r{10}\lt r\lt2r

and

\dfrac r{10}\lt\sqrt r

because

r\lt100.

Also

r^2\lt9\lt10,

so

\dfrac r{10}\lt\dfrac1r.

Thus

\frac{r}{10}

is the smallest of the five quantities, and equation (B) has the largest solution.

Thus, the correct answer is B.

Problem 15 in other years