Video Transcript
A 23-foot ladder leans against the building such that the angle between the ground and the ladder is 80 degrees. How high does the ladder reach up the side of the building? Give your answer to two decimal places.
In this question, we’re given some information about a ladder leaning up against the building. We need to determine how high up the ladder leans against the building. And in questions like this, it’s always a good idea to sketch the information we’re given. First, we’ll have our ladder leaning up against the wall of a building and we’ll also draw in the ground. And we can make an assumption here. We can assume the ground and the wall of the building meet at right angles. Next, we’re told in the question the ladder is 23 feet long, so we can add this to our diagram. Finally, we’re told that the angle between the ground and the ladder is 80 degrees. And we want to determine how high up the ladder reaches the side of the building. We can call this value 𝑥.
Clearing up the information, we can see that we have a right triangle where we know one of the non-right angles and we also know one of the side lengths, and we need to determine another side length. We can then recall we can do this by using right-triangle trigonometry. To do this, we need to start by labeling the sides of our triangle. Let’s start with the hypotenuse. That’s the longest length of the right triangle, which is the one opposite the right angle. In our case, we can see that’s the ladder, which has length 23 feet. Next, we can see that the length of the side opposite the angle of 80 degrees is 𝑥. And remember, that’s the height that the ladder leans up the building. We’ll call this the opposite side.
And the final side of the triangle is the one adjacent to our angle of 80 degrees. So we’ll label this side as the adjacent side. It’s the distance between the base of the building and the base of the ladder. We’re now ready to try and apply right-triangle trigonometry to determine the value of 𝑥. To do this, we’ll start by recalling the acronym SOHCAHTOA. This can be used to help us determine which trigonometric ratio we need to use. To do this, we’re going to need to use our diagram. We can see that we know the length of the hypotenuse, and we want to determine the length of the opposite side to our angle. And by using our acronym, we can see that the sine function relates the opposite side to the hypotenuse in a right triangle.
We can then recall if 𝜃 is an angle in a right triangle, then the sin of 𝜃 is equal to the length of the side opposite angle 𝜃 divided by the length of the hypotenuse. We can then substitute the values we have in our right triangle. Our angle is 80 degrees, the side opposite 80 degrees has a length of 𝑥, and the hypotenuse has a length of 23 feet. sin of 80 degrees is equal to 𝑥 divided by 23. And now we could solve for 𝑥 by multiplying through by 23. 𝑥 is 23 sin of 80 degrees. And we can evaluate this by using our calculator, where we need to be careful to make sure it’s set to degrees mode.
We get that 𝑥 is 22.650 and this expansion continues. And don’t forget, we know this represents a length and these lengths are given in feet. Finally, remember, the question wants us to give our answer to two decimal places. So we look at the third decimal digit, which is zero, which is less than five. So we need to round this value down. This gives us that 𝑥 is 22.65 feet to two decimal places, which is our final answer.
Therefore, we were able to show if a 23-feet ladder leans up against a building such that the angle it makes with the ground is 80 degrees, then to two decimal places, the ladder reaches 22.65 feet up the side of the building.
