Answer
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Hint: Use the relation between apparent angle, true angle and angle of dip with meridian. Substitute the given values directly into the formula and find the true angle of dip.
Formula used:
$\cot ^{ 2 }{ \theta \quad =\quad \cot ^{ 2 }{ { \theta }_{ 1 }+\quad \cot ^{ 2 }{ { \theta }_{ 2 } } } }$
Complete answer:
Let $ { \theta }_{ 1 }$ be the apparent angle of dip.
$ { \theta }_{ 2 }$ be the angle of dip with magnetic meridian.
$ \theta$ be the true angle of dip.
Given: ${ \theta }_{ 2 }$= 30° and
${ \theta }_{ 1 }$= 40°
Now, we have the relation between apparent angle of dip and true angle of dip,
$\cot ^{ 2 }{ \theta \quad =\quad \cot ^{ 2 }{ { \theta }_{ 1 }+\quad \cot ^{ 2 }{ { \theta }_{ 2 } } } }$
$\Rightarrow \cot { \theta \quad =\sqrt { \cot ^{ 2 }{ { \theta }_{ 1 }+\cot ^{ 2 }{ { \theta }_{ 2 } } } } }$
By substituting the values in above equation we get,
$ \cot { \theta \quad =\sqrt { \cot ^{ 2 }{ 40°+ } \cot ^{ 2 }{ 30° } } }$
$\Rightarrow \cot { \theta \quad =\sqrt { { 1.19 }^{ 2 }+{ \sqrt { 3 } }^{ 2 } } }$
$\Rightarrow \cot { \theta \quad =\sqrt { 1.42+3 } }$
$\Rightarrow \cot { \theta \quad =\quad 4.42 }$
$\Rightarrow \quad \theta =\quad 25°$
Therefore, the true angle of dip is less than 40°.
So, the correct answer is “Option D”.
Note:
There’s an alternate method to solve this problem. The alternate method is shown below.
Formula used to find the true dip is given by,
$\tan { { \theta }_{ 1 }=\dfrac { \tan { \theta } }{ cos{ \theta }_{ 2 } } }$
By substituting the values in above equation we get,
$\tan { 40°=\dfrac { \tan { \theta } }{ cos30° } }$
$\Rightarrow \tan { \theta =\quad \tan { 40° } \times \cos { 30° } }$
$\Rightarrow \tan { \theta =\quad 0.84\quad \times \quad 0.87 }$
$\Rightarrow \quad \theta =\tan ^{ -1 }{ 0.73 }$
$\Rightarrow \quad \theta =\quad 36.13°$
Therefore, the true dip is less than 40°.
Hence, the correct answer is option D i.e. less than 40°.
Formula used:
$\cot ^{ 2 }{ \theta \quad =\quad \cot ^{ 2 }{ { \theta }_{ 1 }+\quad \cot ^{ 2 }{ { \theta }_{ 2 } } } }$
Complete answer:
Let $ { \theta }_{ 1 }$ be the apparent angle of dip.
$ { \theta }_{ 2 }$ be the angle of dip with magnetic meridian.
$ \theta$ be the true angle of dip.
Given: ${ \theta }_{ 2 }$= 30° and
${ \theta }_{ 1 }$= 40°
Now, we have the relation between apparent angle of dip and true angle of dip,
$\cot ^{ 2 }{ \theta \quad =\quad \cot ^{ 2 }{ { \theta }_{ 1 }+\quad \cot ^{ 2 }{ { \theta }_{ 2 } } } }$
$\Rightarrow \cot { \theta \quad =\sqrt { \cot ^{ 2 }{ { \theta }_{ 1 }+\cot ^{ 2 }{ { \theta }_{ 2 } } } } }$
By substituting the values in above equation we get,
$ \cot { \theta \quad =\sqrt { \cot ^{ 2 }{ 40°+ } \cot ^{ 2 }{ 30° } } }$
$\Rightarrow \cot { \theta \quad =\sqrt { { 1.19 }^{ 2 }+{ \sqrt { 3 } }^{ 2 } } }$
$\Rightarrow \cot { \theta \quad =\sqrt { 1.42+3 } }$
$\Rightarrow \cot { \theta \quad =\quad 4.42 }$
$\Rightarrow \quad \theta =\quad 25°$
Therefore, the true angle of dip is less than 40°.
So, the correct answer is “Option D”.
Note:
There’s an alternate method to solve this problem. The alternate method is shown below.
Formula used to find the true dip is given by,
$\tan { { \theta }_{ 1 }=\dfrac { \tan { \theta } }{ cos{ \theta }_{ 2 } } }$
By substituting the values in above equation we get,
$\tan { 40°=\dfrac { \tan { \theta } }{ cos30° } }$
$\Rightarrow \tan { \theta =\quad \tan { 40° } \times \cos { 30° } }$
$\Rightarrow \tan { \theta =\quad 0.84\quad \times \quad 0.87 }$
$\Rightarrow \quad \theta =\tan ^{ -1 }{ 0.73 }$
$\Rightarrow \quad \theta =\quad 36.13°$
Therefore, the true dip is less than 40°.
Hence, the correct answer is option D i.e. less than 40°.
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