Cowculus 4
The Crusader Letterbox


Terrain: Kettles - 'nough said

Difficulty: Need to know Calculus...
Created by: The Guys of KML's 10-11 AP Calculus classes

Placed by: The Dragon (Mathman)

Location: West Bend, Washington County

Dragon's Home Page

Mathman Home Page

This box is the fourth in a series of letterboxes with Calculus clues.  Thanks to Z-fan for the idea...apologies to Leapin' Lizards and other math-phobics :-)

Students and parents of students should read my introduction to letterboxing before seeking the boxes.

Solve the following Calculus problems, then enter the answers into the following narrative to find the box.
1)  The oncoming horde's position is given by s(t) = 6t3 + 2t2 + 12t + 8 where t is in days
     and s is in miles.  What is the rate at which the horde's velocity changes at t = 2? A = ______
2)  During the Crusades, catapults were a very popular weapon.  Suppose that a soldier
     launches a stone out of a catapult from atop a tower, and that the stone's path follows
     the function h(t) = -2t2 + 16t + 26.  What is the maximum height of the stone?  B = ______
3) Evaluate the following integral expression. 
    Round the answer to the nearest whole number: C = ______

4)  Evaluate the following integral. Round to the nearest whole number. D = ______

5)  Find the area bounded by y = -x2 + 12x - 20 and y = 1. 
     Round your answer to the nearest whole number.  E = _______
6) Find the area of the region bounded by  , x = 1, and the x-axis.
    Round to the nearest whole number.  F = ______
7)  Find the area enclosed by the polar graph r = 3 + 5cosƟ. 
     Multiply this area by 2 and round to the nearest multiple of ten.
G = ______
To get in the mood, I recommend listening to Chris DeBurgh's song "Crusader" on Groveshark or other online medium.
"Jerusalem is lost!  Jerusalem is lost!" you hear the cry from the priests in the belfries.  "The troops are mustering at Lac Lawrann!  All able bodies must join them there by order of the king!" declare the king's hearalds.  You grab your sword and shield and your marching gear and head to the marshalling grounds.

As you join the troops, you find divisions of troops assembling in the various fields.  You learn that there will be a multi-pronged attack, and you will be in charge of the force
 flanking the enemy in a surprise attack.  You head over to the barn to collect your troops and lead them on their way.

As you head out, you stop to pay respect to Saint Lawrence - a veteran of a previous crusade whom you admire. As you head down the hill to the lake below, you observe the "ring of fire" - a place to test your mettle as a warrior.  Standing in the middle of the ring, you are tested by the flames and overcome the trial.  As a reward, you are given a vision that you should head at a bearing of C = ______ degrees, so you lead the troops off in that direction.

Along the way, you pass the resting place of Sir Robin's minstrels, which leads to much rejoicing.  The rejoicing ends quickly though, as the trail suddenly takes a turn for the worse.  You can see the path that lies ahead of you, but you are forced to lead your troops through the brush to avoid the ravages of nature and war that block your way.  Back on the straight and narrow path, you plod on to the next intersection.

Now begins the long and perilous journey through unfamiliar lands to the Holy Land.  You head off at  F = ______ degrees and soon are diverted to head at D = ______ degrees as your path is split in two.  It is your hope that you can take the enemy by surprise by travelling on higher ground.  But travelling on this trail is not easy.  Another path beckons.  Which should you choose?

You choose to take the path at A = ______.  As you progress, you see many fallen giants from past battles and wonder whether you've chosen wisely.  The forest around you grows darker.  The trail twists and winds. Then it ends suddenly with another path cutting across in front of you.  One way offers a challenge, the other way offers ease.  As a crusader, you understand you must face challenges along the way, so you take up the challenge.  At the crest however, you look down and find that the trail becomes a slippery slope down into a marshy wasteland.  You shudder thinking of the stories from the past telling of soldiers dying gruesomely in that pismire from swamp monsters.  There is no hope of getting through that to your target!  You opt to turn back and take the other path.  As you turn to go back however, you notice the large giant on the side of the trail.  The curve embedded in its center reminds you of the Fibonacci sequence, Archemedes, and conch shells.  You take it as a sign from heaven and vow to remember the number of "heads" _____ on the giant!

You head away from the giant back to your previous location and take the path yet untaken.  To make up time, you choose to head at B = _______ degrees at your next option.  Along the easier path you surge ahead.  The light through the trees tells you you are nearing the edge of the forest, and with it the citadel you seek to capture!  At the next intersection, you hear sounds of battle off at G = _______ degrees, and lead your troops in that direction.  The trail gets bumpy and you tackle many earthen abutments, when suddenly the ground upon which you are travelling becomes softer and turns to sand.  You are now out in the open and the battle for the citadel is going on in full force ahead of you!  Charging up the dune with your forces behind you, you are suddenly blocked by a line of barb-arians.  Rats!  They anticipated an attack from this side.  You make a strategic retreat along the open path to the right looking for a breach in their de-fences.  As you emerge, you meet other troops that have come by other routes, and you discover a break in the enemy's lines at E = ______ degrees!

Charge!!!!!! You sweep down through the enemy hordes and blast your way straight through the hole in the barb-arian's de-fences.  You are in the city!  You take a turn to the left at 10 degrees and lead your troops along the birch-wood ramparts.  Surging straight ahead, you can see the citadel tower looming in the distance ahead of you.  The way is now clear - the victory yours - the enemy is routed.

Wearily, you head over to the tower.  It lies in ruins - a mere shell of what is once was.  Inside you find evidence of other crusaders that came before you.  In particular, you make note of the year Sir Stephen led his troops here.  Putting together the first and last numbers from this date along with the number of heads on the giant, you sum up your victory.  From the northeast outer corner of the tower, you head off in your new-found direction and find your prize under the rusting shield among the debris of battles over long ago.

Before you set out read the waiver of responsibility and disclaimer.

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