Question

The shadow of a tower on level ground is $30\ \text{m}$ longer when the sun's altitude is $30^\circ$ than when it is $60^\circ$. Find the height of the tower. (Use $\sqrt{3}=1.732$.)
 

Hint:

For sun–shadow problems, use $\,\text{shadow}=h\cot\alpha\,$ and the given difference to form a linear equation in $h$.

Answer 1

Let the height of the tower be $h$. For an altitude angle $\alpha$, the shadow length is $h\cot\alpha$.
Shadow at $30^\circ$: $h\cot30^\circ=h\sqrt{3}$.
Shadow at $60^\circ$: $h\cot60^\circ=\dfrac{h}{\sqrt{3}}$.
Given: $h\sqrt{3}-\dfrac{h}{\sqrt{3}}=30 $$\Rightarrow$$ h\left(\dfrac{3-1}{\sqrt{3}}\right)=30 $$\Rightarrow$$ h\cdot\dfrac{2}{\sqrt{3}}=30$.
$\Rightarrow\ h=15\sqrt{3}\ \text{m}=15(1.732)\ \text{m}=25.98\ \text{m}\ (\approx 26\ \text{m}).$
\[ \boxed{h=15\sqrt{3}\ \text{m}\ \approx\ 26\ \text{m}} \]