This problem has difficulty distinguishing between a correct answer and an incorrect answer that is close numerically. Often, this is not a big issue, but here the incorrect answer that many students get reflects a serious conceptual error.

Let me quote an e-mail from my colleague, Margaret Symington. I don't know that there is an easy way to resolve this issue without just re-writing the whole problem. I would appreciate any good ideas.

"The problem is:

The axis of a light in a lighthouse is tilted. When the light points east, it

is inclined upward at x degree(s). When it points north, it is inclined upward

at y degree(s). What is its maximum angle of elevation?

If x and y are the angles of inclination that are given, a "cheap" answer is:

sqrt(x^2+y^2).

While that has some merit in that it stems from recalling how a gradient is

built out of its component directional derivatives, the component directional derivatives are not x and y, but tan(x) and tan(y)... and we have to convert

from degrees to radians and back to degrees. So the correct answer is

(180/pi)*arctan(sqrt((tan(x*pi/180))^2+(tan(y*pi/180))^2)).

But for small angles (between 1 and 8 degrees) that are not close to being

equal, these two functions of x and y are very close. To see what was going

on I plotted the two functions on three different domains. On the smallest

domain you can barely distinguish the two near x=y=10."

## WeBWorK Problems

### multivariate problem: setVmultivariable6Gradient/ur_vc_6_17.pg

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