Answer
Verified
333.5k+ views
Hint: Draw the diagram of the given problem statement for a better understanding of the situation. Use the trigonometric ratios, that are $\sin \theta = \dfrac{{{\text{Opposite}}}}{{{\text{Hypotenuse}}}}$ and $\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$ in the physical triangle formed to find the height of the tower and horizontal distance between the building and the tower.
Complete step by step answer:
Since it is given that the angle observed from the top of the tower to the top and bottom of the building are \[30^\circ \] and \[60^\circ \]respectively, we can draw a diagram representing the condition.
In the diagram, A represents the top of the tower, AB represents the height of the tower, DC represents the building of height 50 metres.
It is given in the question that the angle $\angle TAD$ is \[30^\circ \] and the angle $\angle TAC$ is \[60^\circ \].
By the property of the corresponding angles of the parallel lines, we can say that the angle $\angle ADE$is \[30^\circ \] and the angle $\angle ACB$ is \[60^\circ \].
In the triangle $ADE$, we can say
\[\tan {30^ \circ } = \dfrac{{{\text{AE}}}}{{DE}}\]
On further simplifying
$
\dfrac{1}{{\sqrt 3 }} = \dfrac{{{\text{AE}}}}{{DE}} \\
\sqrt 3 {\text{AE = DE}} \\
$
Similarly, in the triangle ${\text{ACB}}$
\[\tan {60^ \circ } = \dfrac{{{\text{AB}}}}{{{\text{BC}}}}\]
On further simplifying
$
\sqrt 3 = \dfrac{{{\text{AB}}}}{{{\text{BC}}}} \\
\sqrt 3 {\text{BC = AB}} \\
$
From the figure we infer that,
${\text{BC = DE}}$
$
{\text{AB = AE + EB}} \\
{\text{EB = CD = 50}} \\
{\text{AB = AE + 50}} \\
$
Substituting the value ${\text{AE + 50}}$ for ${\text{AB}}$ and ${\text{DE}}$ for ${\text{BC}}$ in the equation \[\sqrt 3 {\text{BC = AB}}\], we get
\[\sqrt 3 {\text{DE = AE + 50}}\]
Also, \[\sqrt 3 {\text{AE = DE}}\]
Thus the expression becomes \[\sqrt 3 \left( {\sqrt 3 {\text{AE}}} \right){\text{ = AE + 50}}\]
We can solve the expression to find the value of ${\text{AE}}$
$
3{\text{AE = AE + 50}} \\
{\text{2AE = 50}} \\
{\text{AE = 25}} \\
$
Substituting the value 25 for ${\text{AE}}$ in the equation ${\text{AB = AE + 50}}$ to find the height of the tower, and in the equation \[\sqrt 3 {\text{AE = DE}}\] to find the horizontal distance between the building and the tower.
$
{\text{AB = 25 + 50}} \\
{\text{AB = 75}} \\
$
\[{\text{DE}} = \sqrt 3 \left( {25} \right)\]
Thus the height of the tower is 75 m, and the horizontal distance between the building and the tower is $25\sqrt 3 $m.
Note: In a right angled triangle, the $\tan \theta $ is the equal to $\dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$, where perpendicular is the side opposite to the angle $\theta $, and $\sin \theta $ is the equal to $\dfrac{{{\text{Perpendicular}}}}{{{\text{Hypotenuse}}}}$, where perpendicular is the side opposite to the angle $\theta $.
Complete step by step answer:
Since it is given that the angle observed from the top of the tower to the top and bottom of the building are \[30^\circ \] and \[60^\circ \]respectively, we can draw a diagram representing the condition.
In the diagram, A represents the top of the tower, AB represents the height of the tower, DC represents the building of height 50 metres.
It is given in the question that the angle $\angle TAD$ is \[30^\circ \] and the angle $\angle TAC$ is \[60^\circ \].
By the property of the corresponding angles of the parallel lines, we can say that the angle $\angle ADE$is \[30^\circ \] and the angle $\angle ACB$ is \[60^\circ \].
In the triangle $ADE$, we can say
\[\tan {30^ \circ } = \dfrac{{{\text{AE}}}}{{DE}}\]
On further simplifying
$
\dfrac{1}{{\sqrt 3 }} = \dfrac{{{\text{AE}}}}{{DE}} \\
\sqrt 3 {\text{AE = DE}} \\
$
Similarly, in the triangle ${\text{ACB}}$
\[\tan {60^ \circ } = \dfrac{{{\text{AB}}}}{{{\text{BC}}}}\]
On further simplifying
$
\sqrt 3 = \dfrac{{{\text{AB}}}}{{{\text{BC}}}} \\
\sqrt 3 {\text{BC = AB}} \\
$
From the figure we infer that,
${\text{BC = DE}}$
$
{\text{AB = AE + EB}} \\
{\text{EB = CD = 50}} \\
{\text{AB = AE + 50}} \\
$
Substituting the value ${\text{AE + 50}}$ for ${\text{AB}}$ and ${\text{DE}}$ for ${\text{BC}}$ in the equation \[\sqrt 3 {\text{BC = AB}}\], we get
\[\sqrt 3 {\text{DE = AE + 50}}\]
Also, \[\sqrt 3 {\text{AE = DE}}\]
Thus the expression becomes \[\sqrt 3 \left( {\sqrt 3 {\text{AE}}} \right){\text{ = AE + 50}}\]
We can solve the expression to find the value of ${\text{AE}}$
$
3{\text{AE = AE + 50}} \\
{\text{2AE = 50}} \\
{\text{AE = 25}} \\
$
Substituting the value 25 for ${\text{AE}}$ in the equation ${\text{AB = AE + 50}}$ to find the height of the tower, and in the equation \[\sqrt 3 {\text{AE = DE}}\] to find the horizontal distance between the building and the tower.
$
{\text{AB = 25 + 50}} \\
{\text{AB = 75}} \\
$
\[{\text{DE}} = \sqrt 3 \left( {25} \right)\]
Thus the height of the tower is 75 m, and the horizontal distance between the building and the tower is $25\sqrt 3 $m.
Note: In a right angled triangle, the $\tan \theta $ is the equal to $\dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$, where perpendicular is the side opposite to the angle $\theta $, and $\sin \theta $ is the equal to $\dfrac{{{\text{Perpendicular}}}}{{{\text{Hypotenuse}}}}$, where perpendicular is the side opposite to the angle $\theta $.
Recently Updated Pages
A hollow sphere of mass M and radius R is rotating class 1 physics JEE_Main
Two radioactive nuclei P and Q in a given sample decay class 1 physics JEE_Main
Let f be a twice differentiable such that fleft x rightfleft class 11 maths JEE_Main
Find the points of intersection of the tangents at class 11 maths JEE_Main
For the two circles x2+y216 and x2+y22y0 there isare class 11 maths JEE_Main
IfFxdfrac1x2intlimits4xleft 4t22Ft rightdt then F4-class-12-maths-JEE_Main
Other Pages
Normality of 03 M phosphorus acid H3PO3 is A 05 B 06 class 11 chemistry JEE_Main
Two charges 9e and 3e are placed at a separation r class 12 physics JEE_Main
The mole fraction of the solute in a 1 molal aqueous class 11 chemistry JEE_Main
The cell in the circuit shown in the figure is ideal class 12 physics JEE_Main
Dissolving 120g of urea molwt60 in 1000g of water gave class 11 chemistry JEE_Main
The nitride ion in lithium nitride is composed of A class 11 chemistry JEE_Main