Question Video: Finding the Height at Which a Ladder Touches a Wall Mathematics

A ladder leans against a wall making an angle of 15° with the wall. The base of the ladder is 0.5 m from the base of the wall. How far up the wall does the ladder reach?

03:13

Video Transcript

A ladder leans against a wall making an angle of 15 degrees with the wall. The base of the ladder is 0.5 meters from the base of the wall. And the question we’re asked is, how far up the wall does the ladder reach?

So with a worded question like this, if I’m not given a diagram, I would always draw my own to start off with. So we’re gonna have a diagram of a ladder, a wall, and a floor. And we’re making the assumption here that the wall is vertical and the floor is horizontal. That seems a reasonable assumption for this question. So here is a sketch of the wall, the floor, and the ladder. Because we assume the wall is vertical and the floor horizontal, we know that we have a right angle here. Now we need to put on the information we’re given. So we’re told the ladder makes an angle of 15 degrees with the wall. So this angle here is 15 degrees. And we’re also told the base of the ladder is 0.5 meters from the base of the wall. So this measurement here is 0.5 meters.

Now what we’re asked to find is how far up the wall the ladder reaches. So we’re being asked to find this measurement here, which I’ll call 𝑦 meters. So I’ve got my diagram. And I can see that actually it’s just a problem about a right-angled triangle. So we’re gonna approach it in exactly the same way as the previous ones. I’m gonna start off by labeling the three sides as always. So I have the hypotenuse, the adjacent, and the opposite. Now let’s recall that tangent ratio that we’re going to need in this question. So I have that tan of the angle 𝜃 is equal to the opposite over the adjacent. You’d be becoming familiar with that by now. So as in the previous questions, I’m gonna write down this ratio again. But I’m gonna fill in the information I know.

So I know that the angle 𝜃 is 15. And I know in this case that the opposite is 0.5. So I have that tan of 15 is equal to 0.5 over 𝑦. Now I need to solve this equation in order to work out the value of 𝑦. So 𝑦 is in the denominator of this fraction. So I’m gonna multiply both sides by 𝑦 in order to bring it up into the top numerator. And when I do that, I have 𝑦 tan 15 is equal to 0.5. Now remember tan 15 is just a number. So I can divide both sides of the equation by it. So I’ll have 𝑦 is equal to 0.5 over tan 15. Now this is the stage where I reach for my calculator in order to evaluate this. And it tells me that 𝑦 is equal to 1.86602 and so on.

Now I need to choose a sensible way to round that answer cause I haven’t been asked for a specific level of accuracy. So the other measurement of 0.5 seems to be given to the nearest tenth. I’ll do the same level of rounding for this value of 𝑦 here. So that will give me that 𝑦 is equal to 1.9. And to answer the question how far up the wall does the ladder reach, I put the units back in. It reaches 1.9 meters up this wall.

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