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A boat sights the top of the lighthouse.

The height of the lighthouse is $40$ foot.

The angle of elevation is ${25^ \circ }$.

Let us draw the diagram as per the above data. It will look as follows:

From the diagram,

At $A$ the boat is present and the $BC$ is the height of the lighthouse.

Consider the $\vartriangle ABC$,

In order to calculate the horizontal distance between the boat and lighthouse we need to calculate the length of $AB$.

Let us apply the $\tan \theta $ in $\vartriangle ABC$ at $A$.

We know the formula of $\tan \theta = \dfrac{{{\text{Opposite}}}}{{Adjacent}}$

So, in $\vartriangle ABC$

$ \Rightarrow \tan A = \dfrac{{BC}}{{AB}}$

Substitute the values $A = {25^ \circ }$and $BC = 40$

$ \Rightarrow \tan {25^ \circ } = \dfrac{{40}}{{AB}}$

Find the value of $AB$ from the above equation.

$ \Rightarrow AB = \dfrac{{40}}{{\tan {{25}^ \circ }}}$

Substitute the value of $\tan {25^ \circ } = 0.46$

If you donâ€™t know the exact value of $\tan {25^ \circ } = 0.46$, you can still get the answer by predicting the approximate value of it from analyzing its graph.

$\

\Rightarrow AB = \dfrac{{40}}{{0.46}} \\

\Rightarrow AB = 86.95 \\

\ $

So, the horizontal distance from the boat to the lighthouse is $87$ foot approximately.