In the given diagram, a 15-foot ladder is leaning against a wall with an angle of elevation of 70 degrees. How high up the wall would it reach? Give your answer to two decimal places.
From the diagram we’ve been given, we can see that the ladder, the wall, and the ground form a right triangle. Let’s begin by adding the information we’ve been given in the question to the diagram. We’re told, first of all, that the ladder is 15-feet long. That’s the length of the line segment joining points 𝐴 and 𝐵. Next, we’re told that the angle of elevation is 70 degrees. Now, an angle of elevation is the angle formed between the horizontal and the line of sight when we look up towards something. So if we stand at the base of the ladder and look up towards the top, then the angle between the horizontal and the line of sight is this angle here. So we can add the value of 70 degrees to our diagram.
We are asked how high up the wall the ladder reaches. We need to calculate then the length of the line segment 𝐵𝐶, which we can label as 𝑥 feet. As we have a right triangle in which we know the length of one side and the measure of one angle and we want to calculate the length of another side, we can approach this problem using right triangle trigonometry. We can label the sides of this triangle in relation to the angle of 70 degrees. The side whose length we want to calculate, 𝐵𝐶, is the opposite. And the side whose length we know, that’s 𝐴𝐵, is the hypotenuse.
Recalling the acronym SOH CAH TOA then, we see that it is the sine ratio that we need to use in this question. This is defined as follows. For an angle 𝜃 in a right triangle, the sin of angle 𝜃 is equal to the length of the opposite side divided by the length of the hypotenuse. We know the value of 𝜃 in this triangle; it’s 70 degrees. And we know the length of the hypotenuse; it’s 15 feet. So we can substitute these values to form an equation. We can also substitute 𝑥 to represent the length of the opposite side. And we have the equation sin of 70 degrees is equal to 𝑥 over 15.
We can now solve this equation to determine the value of 𝑥. And we do this by multiplying both sides of the equation by 15, which gives 𝑥 is equal to 15 sin of 70 degrees. All that’s left to do is to evaluate this on a calculator, ensuring that it is in degree mode. This gives 14.0953 continuing. The question specifies that we should give our answer to two decimal places. So as the digit in the third decimal place is a five, we round up to 14.10. Now, this is a sensible answer because the side of 𝑥 feet is one of the shorter sides of this right triangle. So our value should be less than the length of the hypotenuse, which was 15 feet.
By recalling then that an angle of elevation is the angle measured from the horizontal to the line of sight when we look up towards something and then applying right triangle trigonometry, we found that this ladder would reach 14.10 feet up the wall. And that value is to two decimal places.