Question

At a certain time of the day a tree casts its shadow \[12.5\] feet long. If the height of the tree is \[5\] feet, find the height of the other tree that casts its shadow \[20\] feet long at the same time.

Answer 1

Hint:
Here, the shadow of the tree and the actual height of the tree forms a right-angle triangle.
At first, we will find the angle of elevation by using the height and the length of the shadow of the tree. With the help of it we will find the height of the other tree.

Complete step-by-step answer:
It is given that at a certain time of the day a tree casts its shadow \[12.5\]feet long and the actual height of the tree is \[5\]feet.
We have to find the height of the tree that its shadow \[20\]feets long at the same time.
At first, we will find the angle of elevation of the tree when it casts shadow of \[12.5\]feet.
Let us consider it as the first tree.

Let us consider, for the first tree, AB is the tree and BC be the length of the shadow of the tree.
So, we have, \[AB = 5,BC = 12.5\] and \[\angle ACB\] be the angle of elevation.
Here, the shadow of the tree and the actual height of the tree forms a right-angle triangle.
To find the angle of elevation, we will use the formula of the ratio of the sides and angles of a right-angle triangle.
So, \[\Delta ABC\] is the right-angle triangle then, from the relation between the ratio of the sides and the angles we get,
\[\dfrac{{AB}}{{BC}} = \tan \angle C\]
Let us now substitute the values of \[AB = 5,BC = 12.5\] in the above relation therefore, we get,
\[\dfrac{5}{{12.5}} = \tan \angle C\]
Let us solve the above equation to find angle “C” we get, \[\angle C = {\tan ^{ - 1}}\dfrac{5}{{12.5}}\]
Hence by solving the equation we get, \[\angle C = {21.8^ \circ } \approx {22^ \circ }\]
Since the shadow of the other tree occurs at the same time it has the same angle of elevation.
So, the angle of elevation of the second tree is also \[{22^ \circ }\].
Here for the second tree also we use the same relation between the ratio of sides and the angle,
Considering the same triangle drawn above we get the relation as \[AB = BC \times \tan \angle C\]
Where BC is the length of the shadow, AB is the height of the tree and C is the angle of elevation.
Therefore, the height of the second tree is \[AB = 20 \times \tan {22^ \circ }\]
Solving we get, the height of the second tree is \[ = 8.08\] feet
Hence, the height of another tree is \[8.08\]feet.

Note:
For both the trees the angle of elevation will be the same since the time at which the shadow falls for the second tree is the same as the first tree. Here we use the relation between the ratio of sides and angles of a triangle one of the relation is \[\tan \theta = \dfrac{{{\text{opposite side}}}}{{{\text{adjacent side}}}}\].