This problem is very similar to the "monk" problem from class. It is very abstract, but the power of the Intermediate Value Theorem can be make explicit here.

To get started, let's name a few things. Let $w(t)$ be your weight in pounds at time $t$ and let $h(t)$ be your height in inches at time $t$. Also, let's assume that $t=0$ corresponds to the moment of your birth. The problem is asking you to verify that there is a moment in time, call it $c$, such that your weight (in pounds) equals your height (in inches). That is, we need to verify that there exists $c$ such that $w(c)=h(c)$.

We claim that both $h$ and $w$ are continuous (let's try not to get too bogged down in any silly technicalities; they are all resolvable anyway). As with the "monk" problem, the trick is to define a new function that is the difference of $h$ and $w$ (in either order): $f(t)=h(t)-w(t)$. Then $f$ is continuous.

Now, to make this a little less abstract, let's use some plausible numbers. When my older son was born he was 8 pounds and 21 inches tall. However, now at age 4 he is 50 pounds and 48 inches (he's huge for his age!). To keep this simple, assume my son is exactly 4 years old. What is $f(0)$ for my son? What is $f(4)$ for my son? What does the Intermediate Value Theorem tell us?