Today I worked on some of the applied problems from the OpenStax textbook Algebra and Trigonometry. Among them were the two pictured below.
These questions made me uneasy as soon as I started working on them. Clearly, how tall the human is and how close the human stands to the spotlight are not enough to determine the angle of the spotlight. Of course, we are given another pair of measurements. Yet, in the context of the problem, these are really the same measurements. The ratio between the height of the human and the distance of the human from the spotlight must always be the same as that between the height of the shadow and the distance of the wall from the spotlight. Since it’s those ratios that ought to be giving us the angle of the spotlight, I don’t see how the problems can be done.
(If $H$ is the height of the human and $S$ is the height of the shadow, the ratio between $S-H$ and the distance of the human from the wall will be the same as the other two ratios, also. Note the similar triangles in the diagram.)
On the other hand, in the situation illustrated above, the height of the human and the length of the shadow are sufficient to calculate the angle of the sun. This makes me wonder if I’m missing something in the textbook’s problems.
hmmmmmm?
The obvious question is whether there’s a mistake in the problem. Of course there is: the woman is much shorter than the man.
I always remember the year you dressed as a garden gnome for Halloween, wearing the right bright primary colors and with a hugely tall pointed hat. A hat that could be sliced to make conic sections. Perhaps the hat has grown in my memory, but I’m imagining Problem 3, which begins, “An 8 foot tall gnome…”
So I guess you either want to find the angle or to find multiple possible angles that satisfy the constraints of the problem.
I remember the hat. It very well could have been two feet high, making me an eight-foot gnome. My height was part of the humor of the costume, of course.