# The shadow of a tower standing on level ground is found to be 40m longer when the sun's altitude is 30∘ than when it is 60∘. Find the height of the tower.

Grade:12th pass

## 1 Answers

Pawan Prajapati
askIITians Faculty 60782 Points
one year ago
First draw a figure so that you can understand. Label the figure according to the question. Consider one Right-angled triangle and take the tan60∘ and for the next Right-angled triangle take tan30∘. Equate it and find the height. Complete step-by-step answer: Height is a measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50m" or "The height of an airplane in-flight is about 10000m". When the term is used to describe the vertical position (of, e.g., an airplane) from sea level, height is more often called altitude. Furthermore, if the point is attached to the Earth (e.g., a mountain peak), the altitude (height above sea level) is called elevation. Height may indicate the third dimension, the other two being length and width. Height is normal to the plane formed by the length and width. Height is also used as a name for some more abstract definitions. These include: The altitude of a triangle, which is the length from the vertex of a triangle to the line formed by the opposite side. Measurement in a circular segment of the distance from the midpoint of the arc of the circular segment to the midpoint of the line joining the endpoints of the arc. In a rooted tree, the height of a vertex is the length of the longest downward path to a leaf from that vertex. In algebraic number theory, a "height function" is a measurement related to the minimal polynomial of an algebraic number; among other uses in commutative algebra and representation theory. In-ring theory, the height of a prime ideal is the supremum of the lengths of all chains of prime ideals contained in it. Let the tower be AB. When the sun’s altitude is at 60∘, ∠ACB=60∘and the length of the shadow=BC. When the sun’s altitude is at 30∘, ∠ADB=30∘and the length of the shadow=DB. So the length of the shadow is 40m when the angle changes from 60∘ to 30∘. That is CD=40m. Now we have to find the height of the tower i.e. AB. Here we can see that tower is vertical to ground so ∠ABC=90∘. So considering Right-angled triangle ABC, tanC=Side opposite to angleCSide adjacent to angleC=ABCB tan60∘=ABCB 3–√=ABCB CB=AB3–√ ………… (1) Now considering Right-angled triangle ADC, tanD=Side opposite to angleDSide adjacent to angleD=ABDB tan30∘=ABDB 13–√=ABDB DB=3–√AB So we know DC+CB=DB. 40+CB=3–√AB CB=3–√AB−40 ………. (2) From (1) and (2), AB3–√=3–√AB−40 AB=3–√×3–√AB−403–√ AB=3AB−403–√ 403–√=3AB−AB 2AB=403–√ AB=203–√m So here we get the height of the tower AB=203–√m. Note: Carefully read the question. Your concept regarding height should be cleared. You should also know the substitution. Also, you must know to draw a figure first so you can find the answer quickly. Most mistakes are done by students at tan60∘andtan30∘. The students are confused between these two.