Lab 9: Optimization - Lab preparation

Lab Preparation: For your group’s problem,

Redraw the given diagram with all relevant constants and variables labeled.
Write a function expressing the quantity to be minimized or maximized. This may be expressed in terms of several "input" variables.
Pick one "input” variable and determine possible values of that variable covering all configurations for your problem.

A garden plot is to be 30 square feet and shaped in a circular sector with radius \(R\) and angle \(\theta\). Minimize the length of fence required to enclose the garden.

A cone can be made by cutting a sector of angle \(\varphi\) out of a circle and taping together the resulting edges. Maximize the volume of cone.
A swimmer is 200 feet from shore and a lifeguard is 200 feet down the shore from the closest point. She can run 18 ft/s and can swim at a rate of 5 ft/s. Minimize the time it would take to reach the swimmer.

A sign is 15 feet wide, perpendicular to a straight road, and 20 feet from the road. What is the maximum viewing angle from the road?

A hallway that is 8 feet wide meets another hallway 5 feet wide. What is the shortest length from one wall to another that touches the inside corner as shown in the diagram?
A main blood vessel of radius 7 mm branches to a side artery of radius 5 mm at an angle \(\theta\). The artery carries blood to a point 140 mm from the vessel (measured from the closest point on the main vessel which is 500 mm from the start of the vessel. Poiseuille's law states that the resistance \(R\) against the blood flow is proportional to the length \(L\) of a vessel and inversely proportional to the fourth power of the radius \(r\), \(R = k\cdot \frac{L}{r^4} \), where \(k = 0.16\) is a constant determined by the viscosity of the blood. Label the distance that blood would travel through the larger vessel as \(x\) and the distance through the smaller artery as \(d\). Minimize the total resistance along the route which is the sum of the resistance on each segment.

lab09 6