CLEAR Calculus

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Lab 9: Optimization - part 1

Lab: For your group’s problem,

1. Draw a diagram with all relevant constants and variables labeled.
   
   

2.  Write a function expressing the quantity to be minimized/maximized as a function of one other variable. Show all the work required to derive this formula.
   
   

3. State the domain of the function covering all possible configurations for your problem.
   
   

4. Compute the derivative and determine all critical points of the function. 
   
   

5. Justify that your critical point does give a minimum or maximum as requested.

The stories you are about to read about your classmates are true; only the names have been changed to protect the innocent.

1. 

   To relieve stress from doing homework problems, Julie has taken up gardening. She has designed her garden plot in the shape of a circular sector with radius \(R\) and angle \(\theta\). Based on the amount of vegetables she wants to grow, Julie has determined the garden should have an area of 30 square feet. She needs to build an electric fence around the perimeter to keep Joe out of the tomatoes. Find the dimensions \(R\) and \(\theta\) which minimize the length of fence Julie will need to build.

2. 

   Elizabeth is fed up with AJ's jokes and decides to make him wear a dunce cap in calculus class. She starts out with a paper circle of radius 12 inches and Luke suggests making the hat as large as possible for optimal humiliation effect. To do this, Elizabeth cuts out a sector with angle \(\varphi\) and tapes together the resulting edges to form the cone. Find the magnitude of \(\varphi\) so that the volume of the AJ's dunce cap is maximized.

   1.  

3. Dan and Mike finished their Calculus Lab early and are enjoying the afternoon at Horsetooth Reservoir, but soon get into an argument. Dan pushes Mike off of their "Little Mermaid" floaty (LMF) 200 feet from shore and paddles off. The icy cold water has momentarily made Mike forget he is a really good swimmer. Teri is at a point 200 feet down the shore from the point closest to Mike. She can run 18 ft/s and can swim at a rate of 5 ft/s. To what point on the shore should Teri run before diving into the lake if she wants to reach Mike as quickly as possible? Once Mike falls into the water, he can manage to thrash about for exactly 51 seconds. Can Teri reach him in time?

4. Rachel and Mark are on a road trip. Just when they think they can't take being in the car with each other any longer, they see a billboard advertising

   Have your picture taken with the buffalo

   at Cory & June's

   Hexaflexagon Mega-Warehouse

   The sign is 15 feet wide, perpendicular to a straight road, and 20 feet from the road. Rachel decides to snap a picture of the sign to take back to show everyone in calculus class. At what point on the road would Rachel have the best view of the billboard as they drive by? That is, at what point on the road is the angle subtended by the billboard a maximum? Is the pair tempted to take the 30-mile detour for the photo-op?

5. Tim and Miguel went to a fortune-teller at the state fair who tipped them off that their next calculus exam would have a problem involving ladders sliding down a wall. Cindy suggests they should replicate this in real life to gain an advantage on the rest of the class, so they all decide to bring a long ladder into their dorm. Tim further adds that they should use as long of a ladder as possible for an optimal experience. Cindy points out the problem that they must maneuver the ladder through a right-angle turn where the hallway constricts from 8 feet down to 5 feet wide. What is the longest ladder the trio can carry horizontally around the corner?

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6. Sara and Bev are studying for a biology exam, but wishing they could get back to their favorite homework - calculus. This chapter is on the system of blood vessels in the body, which is made up of arteries, arterioles, capillaries, and veins. Sara wonders out loud if there is a reason for the branching patterns she sees in the textbook diagrams. Candice happens to walk by and overhear the conversation. She suggests that the reason might be that the transport of blood from the heart through all organs of the body and back to the heart should be as efficient as possible. She suggests to Sara and Bev that one way this can be done is by having large enough blood vessels to avoid turbulence and small enough blood cells to minimize viscosity. Then Sara suggests they use calculus to derive the angle \(\theta\) for branches in blood vessels such that total resistance to the flow of blood is minimized. She draws the picture below and says they could assume that a main vessel of radius 7 mm runs along a horizontal line. We want to carry blood to a point 500 mm from the start of the main vessel and 140 mm away from the vessel. A side artery, of radius 5 mm, can head for this point branching at any point along the vessel. Bev points out that in order to solve the problem, they would also have to know how the resistance of blood flow is related to the size of the vessel. Fortunately, Candice remembers that they can use Poiseuille’s law for that. Specifically, the resistance \(R\) in the system is proportional to the length \(L\) of the vessel and \(L\) inversely proportional to the fourth power of the radius \(r\). That is, \(R = k \cdot \frac{L}{r^4}\), where \(k = 0.16\) is a constant determined by the viscosity of the blood. Labeling the distance that blood would travel through the larger vessel \(x\) and the distance through the smaller artery \(d\), Sara notes that the total resistance along the route is the sum of the resistance on each segment. What angle \(\theta\) minimizes this resistance?