Question

A tower is $120m$ high. Its shadow is $xm$ shorter when the sun's altitude is ${60}^{\circ }$ than when it was ${45}^{\circ }$. Find $x$ correct to nearest metre

• A. $51m$
• B. $42m$
• C. $26m$
• D. None of the above

Answer 1

The correct option is A $51m$

Let $AB$ be the tower and its shadows be $BD$ and $BC$ corresponding to the angle of elevations ${60}^{\circ }$ and ${45}^{\circ }$ respectively

In $\Delta ABC,$

$\tan {45}^{\circ }=\frac{AB}{BC}$

$\Rightarrow 1=\frac{120}{BC}$

$\Rightarrow BC=120m$

In $\Delta ABD,$

$\tan {60}^{\circ }=\frac{AB}{BD}$

$\Rightarrow \sqrt{3}=\frac{120}{BD}$

$\Rightarrow BD=40\sqrt{3}m$

Now,

$\begin{array}{cc}BC& =BD+CD\\ 120& =40\sqrt{3}+x\\ x& =120-40\sqrt{3}\\ & =40\sqrt{3}(\sqrt{3}-1)\\ & \approx 51m\end{array}$