Linus, the blanket-clutching philosopher of Charlie Brown fame, awaits the Great Pumpkin while lying in the local pumpkin patch. According to legend, the Great Pumpkin rises each Halloween to award the good boys and girls of the world with candy, and he wants to be the first to obtain visual confirmation of this gourd-related miracle.

In order to keep Linus' innocence intact, Lucy rigs up a powerful cannon to shoot a large pumpkin straight up into the sky. If Lucy positions the cannon a safe 250 feet away from Linus's head, the projectile pumpkin travels according to the position equation (where position is in feet and t is in seconds), and Linus follows the pumpkin with his eyes, how quickly is the angle of elevation (in radians per second) of Linus' sight increasing 5 seconds after the pumpkin is fired?

Hint: The angle theta in the illustration represents the angle I'm talking about.

Difficulty:

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Solution

First of all, we need to set up a primary equation containing the variable we're looking for. Since we're trying to find , the primary equation must contain a . Since Linus' eyesight, a segment marking the height of the pumpkin, and a segment connecting his eyes to the cannon forms a right triangle, we use the tangent formula:

where h is the height of the pumpkin. Derive everything with respect to t to get:

Now, we need to plug in all the values we know when t = 5. Start by figuring out what h is at that exact moment by plugging in 5 for t in the position equation:

True, there's no h in our derivative, but we can use that value to calculate (in radians):

Remember, we're working in radians, so if that angle feels funny, don't worry (it equates to about 51 degrees in case you're curious). Great, so now we know what is. Now it's time to calculate dh/dt. This is the rate at which the pumpkin is traveling, i.e. its velocity. Since we have the position equation, we need to find its derivative to calculate the velocity:

Now, plug what you know into our original derivative and solve for :