Dollar Value of Investments Setup

Question

The dollar value of two investments after t years is given by:

• $f(t)=1800\cdot1.055^t$
• $g(t)=9500\cdot1.041^t$

Solve the equation $f(t) = g(t)$.

What does your solution tell you about the investments?

This is what I have so far:

$1800\cdot1.055^t=9500\cdot1.041^t\implies$

$1800/9500=1.055/1.041$

Am I setting this up correctly so far?

Answer 1

Hint:

$$1800\cdot1.055^t=9500\cdot1.041^t$$

$$\left(\frac{1.055}{1.041}\right)^t=\frac{9500}{1800}$$

Take logarithm and solve for $t$.

Answer 2

You have $1800(1.055)^t=9500(1.041)^t$

Dividing both sides by $1800$ and $1.041^t$

$\frac{1.055^t}{1.041^t}=\frac{9500}{1800}$

Here you have mixed something up.

$\left( \frac{1.055}{1.041}\right)^t=\frac{9500}{1800}$

Taking logs

$\log \left[ \left( \frac{1.055}{1.041}\right)^t\right]=\log\left( \frac{9500}{1800}\right)$

$t\cdot \log \left[ \frac{1.055}{1.041}\right]=\log\left( \frac{9500}{1800}\right)$

Thus $$t=\frac{\log\left( \frac{9500}{1800}\right)}{\log \left[ \frac{1.055}{1.041}\right]}=124.523\approx 125$$