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A Fright for Halloween!

Happy Halloween everyone.  Last year, I posted this Calculus Halloween test. I thought it was a lot of fun, so I wanted to post it again for your enjoyment.

If you get the answers, post them in the comments! Hopefully we can work together to stay out of harm’s way.

A witch’s delite

Suppose that a witch is flying around town and plucking up all of the children in town to bring back to her home. Before cooking she places each child in a cage. The cages are marked too skinny and nice and plump. Is the witch’s mapping of the town’s children to cages a function? If so, is it invertible?

Run! It’s a werewolf!

You are standing in a field harvesting pumpkins on one side of a river that is 1000ft wide. On the other side, your house is 5000 ft down the water bank. However, 7200ft in the other direction you spot a werewolf running toward your house. While your door is reinforced against werewolf attack, your children had left it open. If you can make it to your house before the werewolf, you can close the door and save everyone. Otherwise… The werewolf can run 1000 feet per minute, while you can only run 500 feet per minute. Furthermore, you can only swim at 350 feet per minute. Can you beat the werewolf back to your house? If so, how would you accomplish this?

Unraveling a mummy

You are visiting a crypt and there is a casket against the wall. On the top of the casket reads a sign, “Open the casket for a surprise!” You, of course, open the casket, and a mummy lunges toward you. You turn and run away as the mummy chases after you. Upon leaving the casket, the mummy’s wrapping from his foot are stuck to the top of the casket. As he walks, the wrapping continue to unravel, being pulled through the bottom of his foot into a taught line to the top of the casket. If the casket if 6ft tall, and the mummy is walking 2 feet per second, how fast are the wrappings unraveling?

Virus Outbreak!

ALERT! There has been a virus outbreak. The virus turns a person into a zombie and is spread when the zombie bites a person! If there are currently 100 zombies, we can model the population of zombies based on the assumption that the growth rate is proportional to the product of the number of zombies and the initial number of humans available to turn into zombies. If there are currently 7 billion people, time is measured in months and the constant of proportionality is .1, determine the number of zombies as a function of time. How long will it be before there are more zombies than people?

Yumm!

After a vampire feeds, the amount of energy felt by the vampire as a function of time is given by
\begin{align*}
\frac{1}{3}t^{3}-\frac{7}{2}t^{2}+6t
\end{align*}
where \(t\) is in hours and energy is in blood energy units. What is the maximum and minimum energy felt by the vampire between the time it feeds and eight hours later?

Watch out for the angry mob

An angry mob has headed out in search of Frankenstein’s monster. They find him and begin to attack him; however, the monster fights back against the mob. The mob damages the monster at a rate of \(10\) damage per minute until the monster has taken a total of 100 damage. As the monster is damaged, he becomes more violent with his attacks. He fights back against the mob with a damage rate of \((10-\frac{h}{10})^{2}\) damage per minute where \(h\) is his remaining health. If each person can take 50 damage before dying, how many people will the monster kill?

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