Question

The length of the shadow of a tower standing on the level plane is found to be $2x$ metres longer when the sun’s altitude is 30 than when it was 45, then the height of tower $x\left( {\sqrt 3 + 2} \right)$, that is true/false
1) True
2) False

Answer 1

Hint: We will first draw the figure corresponding to the given condition. Two triangles will be formed. Form equations using both the triangles. Then substitute the value of unknowns in the equations to find the height of the tower. If the answer matches with the value given in the question, the answer is true otherwise it is false.

Complete step by step solution:
In these types of questions, we begin by drawing the figure from the given conditions.

The height of the tower is $h$metres. The shadow of the tower is represented by $DC$and is of length $2x$ meters.
Let the distance from the base of the tower to the shadow be $y$metres represented by $BC$.
We will find the value of $y$from $\Delta ABC$
Now we will calculate the height of the tower when the sun’s altitude is ${45^ \circ }$.
Find the value of $\tan {45^ \circ }$ in $\Delta ABC$using the formula \[\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}\]
Hence, $\tan {45^ \circ } = \dfrac{{AB}}{{AC}}$
Also,\[\tan {45^ \circ } = 1\] therefore, on substituting the values of $AB$and $BC$, we get
$1 = \dfrac{h}{y}$
$h = y{\text{ }}\left( 1 \right)$
Now we will calculate the height of the tower, when the sun’s altitude is ${30^ \circ }$.
Find the value of $\tan {30^ \circ }$ in $\Delta ABD$ using the formula \[\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}\]
Hence, $\tan {30^ \circ } = \dfrac{{AB}}{{BD}}$
Also,\[\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}\] therefore, on substituting the values of $AB$and $BD$, we get
$\dfrac{1}{{\sqrt 3 }} = \dfrac{h}{{2x + y}}$
Height can be written as, \[\sqrt 3 h = 2x + y\].
After we substitute the value of \[y\]from equation (1), we get,
$
  \sqrt 3 h = 2x + h \\
  h\left( {\sqrt 3 - 1} \right) = 2 \\
  h = \dfrac{{2x}}{{\left( {{\kern 1pt} \sqrt 3 - 1} \right)}} \\
$
Since the denominator of the fraction is not a rational number, we will rationalise it.
 $
  h = \dfrac{{2x\left( {\sqrt 3 + 1} \right)}}{{\left( {{\kern 1pt} \sqrt 3 - 1} \right)\left( {\sqrt 3 + 1} \right)}} \\
   = \dfrac{{2x\left( {\sqrt 3 + 1} \right)}}{{3 - 1}} \\
   = \dfrac{{2x\left( {\sqrt 3 + 1} \right)}}{2} \\
   = x\left( {\sqrt 3 + 1} \right) \\
$
Therefore, the height of the tower is \[x\left( {\sqrt 3 + 1} \right)\] m.
Hence, the given statement that the length of the shadow of a tower standing on the level plane is found to be $2x$ metres longer when the sun’s altitude is 30 than when it was 45, then the height of tower $x\left( {\sqrt 3 + 2} \right)$m is false.

Note: Students make mistakes in marking angles in the figure, which gives the wrong answer. This question can alternatively be done by putting $BC$ equals to $h$ as it is given that the length of the shadow of a tower standing on the level plane is found to be $2x$ metres longer.