A tower casts a shadow $90 m$ long and at the same time another tower casts a shadow of $120 m$ on the ground. If the height of the second tower is $80 m$ , then the height of the first tower is:

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60 m

5.5 m

50 m

40 m

answer is A.

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Detailed Solution

Given that, the shadow of a tower is

90 m

long and the shadow of second tower is

120 m

on the ground.
Question Image

The Angle-Angle (AA) criterion for similarity of two triangles states that “If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar”.
Since both the towers cast the shadows at the same time, it means that the Sun’s altitude is the same for both the towers.
Now in ΔABC and ΔDEC,
∠ABC=∠DEC=90° (because both towers are standing straight on ground)
∠BAC=∠EDC (because sun rays fall at the equal angle with the tower)
Therefore, by AA similarity,
ΔABC∿ΔDEC
So, the ratio of the heights of the towers to the lengths of their shadows are same.
Let h be the height of the first tower.
Then,