Heights and Distances

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle to start. When the sun is thirty degrees above the horizon, the shadow of a tower is twenty metres longer than when the sun is sixty degrees up. Can you find the height of the tower from just that, with no tape and no climbing? Sounds impossible, right? Today we will see that two angles and one distance are enough. Heights and distances questions do appear in JEE, and they are simple once you pick the right triangle. So, let's begin!

0:31Angles of elevation and depression

So, here is our picture. The thick line P Q is a tower, with P the top, Q the foot, and h its height. The points A and B are on level ground, and the curly brace shows that they are d apart. With me? Stand at the point A and look up at the top. The angle between your line of sight and the horizontal is the angle of elevation. Here it is alpha, and from the point B it is beta; both are Greek letters. Okay? Now flip it. Someone at the top P looks down at the point B on the ground. Let me ask you this: is the angle of depression measured from the tower, or from the horizontal? Yes, always from the horizontal, which is the dashed line through the top. The angle between the line of sight and this dashed line is the angle of depression. Measuring it from the tower is a classic slip. Right? Now read the note at the top of the figure: the angle of depression of B from P equals beta. But why? Look at the dashed line and the ground. Because the dashed line is parallel to the ground, exactly! So these are alternate angles, and alternate angles are equal. The angle of depression of B from P equals the angle of elevation of P from B, okay? Every depression question becomes an elevation question. Got it?

1:54Draw first, then pick the right triangle

Now, let us find h from alpha, beta and d. Many students write straight away that tan alpha equals h over d. Is that correct? No, no, careful, that is the wrong triangle! The side d is A B, and the triangle PAB has no right angle at all. The right angle is at the foot of the tower, so mark it first, then use the right triangles PQB and PQA, okay? The little triangle symbol just means triangle. In triangle PQB, tan beta is opposite over adjacent: h over Q B, the side next to beta. The double arrow on screen means implies. So Q B equals h times cot beta, where cot, the cotangent, is one over tan. Right? In the same way, in the bigger triangle PQA, the side Q A equals h times cot alpha. Clear? On the ground, d is Q A minus Q B, as the figure shows. So d equals h times, cot alpha minus cot beta. With me? Divide, and here is the first standard result. h equals d over, cot alpha minus cot beta. Or, in sines, d times sine alpha times sine beta, over sine of beta minus alpha. Here alpha is less than beta, since B is nearer the tower, okay?

3:19Two more standard results

Now two more standard results. Here one point is due south of the tower, the other due west, or due east, d apart, with elevations alpha and beta. Clear? Now, south and west: what angle do these two directions make at the foot of the tower? It is ninety degrees, correct! So call the foot Q, the south point A, and the west point B, okay? Just as before, Q A equals h times cot alpha, and Q B equals h times cot beta. These two distances are at right angles, and d joins their ends. Right? So by Pythagoras, d squared equals h squared times, cot squared alpha plus cot squared beta. Take the square root, and h equals d over the square root of, cot squared alpha plus cot squared beta. Got it? And now a topper's trick. Two vertical poles have heights small a and small b. Join the top of each pole to the foot of the other, okay? These two lines meet at a height of small a times small b, over small a plus small b, whatever the distance between the poles! Clear?

4:27Example 1: tower height from shadows at 30 and 60 degrees

Now, remember our puzzle? Let's solve it. The shadow of a tower is twenty metres longer when the sun's altitude is thirty degrees than when it is sixty degrees; the small circle means degrees, and m is metres. Find the height. Pause the video and try it first. Done? Okay, let's check. The sun's altitude is the angle of its rays above the horizontal. So in our figure, where are the tips of the two shadows? At A and B, yes, exactly! The ray grazing the tower's top lands at the shadow's tip. So Q B is the short shadow, with beta sixty degrees, and Q A the long one, with alpha thirty degrees. Each shadow is h times cot of the sun's altitude. Okay? The long shadow minus the short one is twenty. So h times cot thirty degrees, minus h times cot sixty degrees, equals twenty. Right? Cot thirty degrees is root three, and cot sixty degrees is one over root three. So h times the bracket, root three minus one over root three, equals twenty. The bracket is two over root three, so h times two over root three equals twenty; the dot means times. With me? So h equals twenty times root three, over two, which is ten root three metres. And that solves our puzzle! Clear? Let us check. Ten root three times cot thirty degrees is thirty metres, and ten root three times cot sixty degrees is ten metres. Thirty minus ten is twenty, so it fits!

6:04Example 2: tower height from depressions off a 60 m cliff

Now a harder one. From the top of a sixty metre cliff, the angles of depression of the top and bottom of a tower are thirty degrees and sixty degrees. Find the height of the tower. Draw the figure, then pause the video and try it. Done? Okay, let's check. First, which angle belongs to the bottom of the tower, thirty or sixty degrees? It is sixty degrees, correct! The bottom is lower, so you look down more steeply, and the top of the tower gets thirty degrees. Okay? Start with the bottom. By alternate angles, the cliff top is seen from the tower's foot at sixty degrees. Call it x, the horizontal distance; the cliff, sixty metres, is opposite to the angle. So x equals sixty times cot sixty degrees, sixty over root three, which is twenty root three metres. Right? Now the thirty degree line, to the tower's top. Many students write that tan thirty degrees equals the tower's height, over x. Is that the right triangle? No, no, wait, that is the wrong triangle again! The thirty degree angle sits at the cliff top, so its vertical side is the drop from the cliff top to the tower's top, not the tower. That drop is x times tan thirty degrees: twenty root three times one over root three, which is twenty metres. With me? So the tower's top is twenty metres below the cliff top, and the tower is sixty minus twenty, which equals forty metres. See? Pick the correct triangle, and it is easy!

7:39Quick revision: elevation, depression, two-point formula

So, quick revision. Elevation and depression are always measured from the horizontal, and by alternate angles, depression equals elevation. Draw the figure first, mark the right angle, and use tan as opposite over adjacent in that triangle. For two points on one side of a tower, d apart, h equals d over, cot alpha minus cot beta. Now go and try the practice problems on this page. All the best!