A ladder of height twenty-four feet leaning against a wall reaches a window
nineteen feet above the ground. To the nearest tenth, how far from the building is the bottom of
the ladder? Here, we have drawn a house with a wall and a ladder leaning against it. So let’s go
ahead and label some things. It says a ladder of height twenty-four feet; that means the ladder
is twenty-four feet tall.
In other words, the length of the ladder’s twenty-four feet. Next it says this ladder
reaches a window that is nineteen feet above the ground. So our ladder touches the wall by the window at nineteen feet. It says, “To the
nearest tenth, how far from the building is the bottom of the ladder?” So the bottom of the ladder is shown and we wanna to know how far from the building
is the bottom of the ladder. We’ll call that 𝑥.
Now we’ve created a triangle with side lengths 𝑥, nineteen, and twenty-four.
Since a house should have walls that are directly straight up and down, meaning they would
be perpendicular to the ground, they should make ninety-degree angles. So in fact, we actually
have a triangle with a ninety-degree angle.
And since we have a ninety-degree angle, we can use the Pythagorean theorem.
And the Pythagorean theorem is 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑎 and 𝑏
are the legs
and 𝑐 is your hypotenuse.
The hypotenuse side is always across from the ninety-degree angle and is the longest side.
Therefore, we can plug in nineteen and 𝑥 for our 𝑎 and 𝑏, the legs, and twenty-four for the
hypotenuse we’ve just seen.
Again, we will replace 𝑎 with nineteen, 𝑏 with 𝑥, and 𝑐 with twenty-four.
Now it actually doesn’t matter if you want to plug nineteen in for 𝑎 or nineteen in for 𝑏 because
addition is commutative. Now we need to square each of these numbers in our variable 𝑥. So nineteen
squared is three hundred and sixty-one,
𝑥 squared is 𝑥 squared or 𝑥 to the second power,
and twenty-four squared is five hundred and seventy-six.
In order to solve for 𝑥, we need to first subtract three hundred and sixty-one from both
sides of the equation.
This makes the three hundred and sixty-ones cancel on the left-hand side and five
hundred and seventy-six minus three hundred and sixty-one is two hundred and fifteen. Now in
order to solve for 𝑥 or isolate 𝑥, to get it all by itself, we need to get rid of the squared.
And the opposite of squaring a number is to square root it because they’re inverse operations.
And we get 𝑥 equals the square of two hundred and fifteen which is
fourteen point six six two. However, we’re supposed to round
to the nearest tenth.
That means we need to round six, so we look to the number to the right,
which is another six, and we decide if that’s going to round six up to seven or keep it
Numbers five through nine will round six up; numbers zero through four will keep six the
same. Therefore, we need to round up.
Therefore, 𝑥 equals fourteen point seven feet. The building and the bottom of the ladder
are fourteen point seven feet apart.