2009 AMC 10B Problem 10

Draw the picture: the stump stands straight up, and the broken top leans from the top of the stump down to the ground. That is a right triangle.

1. The stump is $x$ meters tall. This is one leg.
2. The top touches the ground $1$ meter from the base. This is the other leg.
3. The broken top is the rest of the pole, $5 - x$ meters. It is the slanted side (the hypotenuse).
4. By the Pythagorean theorem:
$$x^2 + 1^2 = (5 - x)^2.$$
5. Expand the right side: $(5 - x)^2 = 25 - 10x + x^2$. So
$$x^2 + 1 = 25 - 10x + x^2.$$
6. Both sides have $x^2$, so take it away from both: $1 = 25 - 10x$. Then $10x = 24$ and $x = 2.4$.
7. Check: the legs are $2.4$ and $1$, the slanted side is $5 - 2.4 = 2.6$. And $2.4^2 + 1 = 5.76 + 1 = 6.76 = 2.6^2$. Correct.

The answer is $\boxed{\textbf{(E)}\ 2.4}$.

In a "snapped pole" problem, one leg plus the slanted side adds up to the known total height. Because $x$ is squared on both sides of the equation, the squares cancel and you only have a simple equation left, with no quadratic formula needed. The triangle $2.4, 1, 2.6$ is a shrunken copy of the $12, 5, 13$ right triangle, which is a nice confirmation.

Test the choices. For each $x$, check whether $x^2 + 1$ equals $(5 - x)^2$. For $x = 2.4$: $5.76 + 1 = 6.76$ and $2.6^2 = 6.76$. For $x = 2.3$: $5.29 + 1 = 6.29$ but $2.7^2 = 7.29$. Only $2.4$ works.

The stump is vertical with height $x$; the upper part has length $5 - x$ and leans from the top of the stump to a point on the ground $1$ meter from the base. These form a right triangle with legs $x$ and $1$ and hypotenuse $5 - x$:
$$x^2 + 1 = (5 - x)^2 = 25 - 10x + x^2.$$
The $x^2$ terms cancel, leaving $1 = 25 - 10x$, so $10x = 24$ and $x = 2.4$.

Check: legs $2.4$ and $1$, hypotenuse $2.6$; indeed $2.4^2 + 1 = 5.76 + 1 = 6.76 = 2.6^2$.

The answer is $\boxed{\textbf{(E)}\ 2.4}$.