Course:MATH180/Archive/2011-2012/103/Some Questions

Here are some questions we will learn how to solve during the term.

Questions 1-2 are about related rates, questions 3-10 concern optimization and question 11-13 Taylor polynomials.

Speed Camera

A police car is staying at 30 meters from a straight road monitoring the speed of the cars passing by. The way the speed camera is used, it measures the speed when a car is at 50 meters of the police car. If a car is getting closer to the police car at 38 km/hour, is it speeding if the speed limit is 50 km/h on that road?

Similar question solved : here
Radar Station
An airplane flies horizontally over a radar station at an altitude of 8000 meters at a speed of 900 km/h. At what speed is it getting farther from the radar station when it is at 10 km from it?
Ships
Some survivors of a shipwreck are on boat raft. At noon, they are 20 km south of a sailboat. Due to the currents, the survivors are traveling east at 20 km/h whereas the sailboat is going south at 40km/h. 1) At what time are the two ships the closest to one another? 2) If the visibility at sea is 10km that day, will the survivors be seen and (hopefully!) rescued?
Biggest Display Area
The front arcade of an Italian-style building has the shape of the curve $y=15-x^{2}$ . The new owner of the building wants to build a shop on the ground floor and in order to attract as many customers as possible he wants to have the biggest display area (front window) as possible. The city's regulations imposes that shops have rectangle front windows. What are the optimal dimensions?
Beam's Resistance
The resistance of a rectangular-shaped wooden beam is directly proportional to its width times the square of its height. What is the strongest beam that can be cut out of a cylindrical piece of wood?
Fence
A farmer has a two-kilometer-long fence to build a rectangular paddock. He wants to give as much space as possible to his horses. So, what are the dimensions that maximises the area for the horses?

For whatever reason, he suddenly wants to minimise this area, what are the new dimensions?
Book Dimensions
The pages of book must have a surface of 560 $cm^{2}$ . The pages have 2.5-centimeters margins on the left, right- and bottom hand sides and a 1-centimeter margin on the top hand side. What dimensions should the pages have so that one can print as much text as possible?
Propane Container
One wants to build a propane container whose shape is a cylinder topped with a hemisphere on each end (like the shape of a pill). The spherical parts being more complicated to build, a square meter of the spherical part costs twice as much as a square meter of the cylindrical part. If one wants to build a container whose volume is $\pi /4~m^{3}$ , what are the dimensions that will minimise the cost?
Pipe-line Path
The company Oyster Inc. wants to build a pipe-line between two cities of Exxon and Mobile that are on the opposite banks of 1-km-wide river. Moreover, the two cities are 3 km apart from each other.

The pipe-line crosses the river from Exxon to a point P on the other bank and the follows the river to Mobile. Knowing that a kilometer crossing the river costs four times more that a kilometer on the bank, what is the path that minimises the cost of this pipe-line.
Conic Cups
a) A company that produces paper conic-shaped cups is asks by an airline to produce cups containing $27\pi ~cm^{3}$ of liquid. What are the dimensions that minimises the surface (and thus the paper used and the cost) of the cups?

b)Same question if the cups have a 'usual' shape (i.e., a cone without top) and if the radius of the base has to be 5 cm to fits in the airline's trays?
Taylor Polynomials
Approximate $e^{3x^{2}}$ by a polynomial. How accurate is the approximation?
Taylor Polynomials (application)
Give an approximation of $\cos(0.25)$ accurate to 7 decimal places.
More Taylor Polynomials (application)
Compute the sixth derivative of $\sin x^{2}$ for x = 0.
Newton's Method
How can we compute ${\sqrt {2}}$ with a precision of decimal places?