Video Transcript
A 23-foot ladder leans against a 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.
We begin by sketching a diagram to model the scenario. We are told that a 23-foot ladder leans against the building and that the angle between the ground and the ladder is 80 degrees. Assuming that the building is perpendicular to the ground, we can create a right triangle as shown. In this right triangle, we know that the length of the hypotenuse is 23 feet. We are trying to work out how high the ladder reaches up the side of the building. This is the side opposite the 80-degree angle.
Recalling the acronym SOH CAH TOA, we know that the sin of any angle 𝜃 is equal to the opposite over the hypotenuse. By letting the height we are trying to calculate be 𝑥 feet, we have sin of 80 degrees is equal to 𝑥 over 23. We can then multiply through by 23 such that 𝑥 is equal to 23 multiplied by the sin of 80 degrees. Ensuring that our calculator is in degree mode, we have 𝑥 is equal to 22.6505 and so on.
We are asked to round our answer to two decimal places. As the third digit after the decimal point is a zero, we round down. And we can therefore conclude that the ladder reaches a height of 22.65 feet up the side of the building.
