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 at six.
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.