A window cleaner has a ladder that is 8.1 meters long. If he places it on the ground such that its top is at a window that is 6.76 meters above the ground, determine the distance between the base of the ladder and the wall and round the result to the nearest hundredth.
Here, we have a house and a ladder. And the ladder is placed on the ground such that the top of the ladder is at the window. We know that the ladder itself is 8.1 meters long. Next, we know something about the window. It says if he places it on the ground such that its top is at a window that is 6.76 meters above the ground. So the window is 6.76 meters above the ground. And it says “determine the distance between the base of the ladder which you see at the bottom and the wall.”
So we want to know the distance between these two points. So we have created a triangle and we’ve actually created a certain kind of triangle. This will be a right triangle because a house or wall and the ground should be perpendicular. If it wouldn’t be, the house will be tilted. So let’s call this distance 𝑥. So here is our triangle again. So since we have a right triangle and we’re finding a missing side length, we can use the Pythagorean theorem to solve for it.
The Pythagorean theorem states the square of the longest side is equal to the sum of the squares of the shorter sides. Across from the 90-degree angle, that side is considered the longest side — also known as the hypotenuse. And the other two sides will be the shorter sides — also known as the legs. So let’s go ahead and plug in.
We’ve plugged in 8.1 for the longest side, 6.76 for the shorter side, and 𝑥 for the shorter side. So to solve for 𝑥, let’s go ahead and square everything. And we get that 8.1 squared is 65.61, 6.76 squared is equal to 45.6976, and then we have 𝑥 squared. So to solve for 𝑥 squared, let’s go ahead and subtract the 45.6976 from both sides of the equation. And we have that 19.9124 is equal to 𝑥 squared.
And now, we need to square root both sides and we get that 𝑥 is approximately 4.46233. However, we’re supposed to round the result to the nearest hundredth and that will be here. So we look to the number to the right of the six, which is a two. So do we keep it a six or round up to a seven? Well, since two is less than five, we will keep this six a six.
Therefore, the distance between the base of the ladder and the wall will be 4.46 meters.