A $10\,\mathrm{m}$ tall tree casts a shadow $10\sqrt{3}\,\mathrm{m}$ long. Calculate the measure of the angle at which the sun’s rays hit the Earth surface.
Students made an auxiliary drawing and denoted the required angle as $\alpha$:
Francis's solution:
He calculated the measure of $\alpha$ using the sine function: $$ \begin{align} \sin\alpha & =\frac{10}{10\sqrt{3}} \cr \sin\alpha &=\frac{1}{\sqrt{3}} \cr \alpha & \approx 35^{\circ} \end{align} $$
Damian's solution:
He first calculated the length of the hypotenuse using the Pythagorean theorem: $$ \begin{align} 10^2+(10\sqrt{3})^2&=c^2 \cr c^2&=400 \cr c&=20 \end{align} $$ Then he determined the measure of $\alpha$: $$ \begin{align} \sin\alpha &= \frac{10\sqrt{3}}{20} \cr \sin\alpha &=\frac{\sqrt{3}}2 \cr \alpha &=30^{\circ} \end{align} $$
Antonina's solution:
She calculated the measure of $\alpha$ using the tangent function: $$ \begin{align} \mathrm{tan}\,\alpha &=\frac{10}{10\sqrt{3}} \cr \mathrm{tan}\,\alpha &=\frac{1}{\sqrt{3}} \cr \alpha &=30^{\circ} \end{align} $$ Which of them solved the task correctly?
Francis
Antonina
Antonina and Damian
None of them. It should have been: $$ \begin{align} \mathrm{tan}\,\alpha &= \frac{10\sqrt{3}}{10} \cr \mathrm{tan}\,\alpha &=\sqrt{3} \cr \alpha& =60^{\circ} \end{align} $$
