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Calculus: Related rates problem.

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A multipurpose shoebox-shaped hall is planned. Its length is supposed to be
x = 50 meters, the width is supposed to be y = 30 meters, and the height is 10
meters. Furthermore, an expensive diagonal beam is planned, hosting a movable
camera car, starting in one (left, back, down) corner and going through the whole
hall until the (right, front, up) corner.

The cost of the base is $50 per square meter, the cost of the roof is $200 per
square meter, and the cost of the sides is $100 per square meter. Furthermore, one
meter length of the diagonal beam costs $ 3000.

a) Seeing the plan, the project leader wants the hall to be longer, maintaining the
height of 10 meters. Since the cost of the whole construction (including the diagonal
beam) is supposed to be constant, the width must be reduced. The question is: If
one increases the length x by 1 cm, how much must the width y be reduced to keep
the cost constant? More formally, the question is to find an equation between the rate
of change dx/dt of the length and the rate of change dy/dt of the width.

b) Can you find such an equation between dx/dt and dy/dt, not just for x = 50 and
y = 30, but for general x and y?

Tools required for this project:
Calculus: Related Rates
Algebra
Geometry: Theorem of Pythagoras

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Solution Preview

Solution:
It will be helpful to first draw a picture of the hall and label the sides.

Determine the total cost of the original hall.
Total Cost = Cost of base + Cost of roof + Cost of 4 sides + Cost of beam
For example, Cost of base = area of base * price per square meter
= x*y*50
=50*30*5
=$75,000

Figure out the ...

Solution Summary

Steps to finding the adjusted dimensions of a shoebox-shaped hall and finding the related rates equation of the length and width of the hall. Includes a pdf attachment.

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Related Rates and calculus problems for real life situations

SEE ATTACHMENT FOR ALL PROBLEM QUESTIONS

Related Rates Problems
1. A tiger escapes from a truck, right in front of the Empire State Building. I start running west along 34th Street at 2.5 m/s, while my friend takes off north on Fifth Avenue at 3 m/s.

Draw a diagram of this situation.

How fast is the distance between my friend and I growing after 12 seconds?

After 12.8 minutes, I've hit the river; I stop to catch my breath. My friend is still running up Fifth Avenue. At what rate is the angle between my friend and I increasing at 13 minutes? How about at 15 minutes? (NB: Convert minutes to seconds before computing)

2. Huckleberry Finn is drifting down the Mississippi on a rectangular, flat- bottomed boat, 6 feet wide by 12 feet long. Suddenly, he hits a rock and gashes a hole in the boat. Water is now pouring into the boat at a rate of 1/2 cubic foot per minute. Huck is doing the best he can to bail out the boat, but he only has a tin cup to pitch the water, and can only remove 1/3 ft^3/min.

Try to picture this dire situation in your mind, and draw a diagram or two to help you understand what's going on here.

At what rate is the volume of water inside the boat increasing?

At what rate is the depth of the water in the boat in increasing?

If the boat is half a foot high, when will the boat completely fill with water?

3. Your esteemed professor has just finished teaching and needs to lift his spirits. He goes to the nearest bar and orders a gin martini, served up in a conical cocktail glass, and downs it. The bartender has filled it to the brim, though, so knowing his clumsiness he decides it's better to drink the first centimeter of it through a straw rather than move the glass and spill any of it. The straw is kind of narrow, so he can only drink at a rate of 10 ml/sec. The cocktail glass is a right angle cone, and it holds 133 ml.

Picture this situation, and then draw a diagram. A delicious, gin-and- vermouth-flavored diagram.

Given that its height and radius are the same, and its volume is 133 ml, what is its height in centimeters? (the volume of a cone is (1 /3)πr2h, where r is the radius and h is the height)

At what rate is the depth of the liquid in the glass decreasing, as a function of time?

How long does it take for the depth of liquid to drop 1 centimeter, so he can safely pick up his glass?

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