Assignment:
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
Provide complete and step by step solution for the question and show calculations and use formulas.