Calculus I Take-Home Problems for Exam III

1.  (See a picture under Section Project, Building A Pipeline, page 920 in the exercises for section 12.9 in your text.--This is a Calculus I problem.)  An oil company wishes to construct a pipeline from its offshore facility A to its refinery B.  The offshore facility is 2 miles from the shore, and the refinery is 1 mile inland.  A and B are 5 miles apart as indicated in the figure in your text.  The cost of building the pipeline is $3 million per mile in the water, and $4 million per mile on land.  The total cost depends on the location of the point P in the textbook figure, P being the point where the pipeline meets the shore.  Let x be as indicated in the figure and determine the value for x that will minimize the total cost and approximate this minimum total cost.  Click here to see my animation for this problem.  In the animation A is at (0,17), B is at (4,14), and the distance between A and B (5 miles) is indicated in black.  Blue indicates the pipeline distance through water and orange the pipeline distance on land.  The distance from the point (0,15) to the animated blue point on the green line is x.  The graph of the total cost function, C(x), is in red and the animated blue point moving along the total cost function corresponds to the animated blue point (P) on the green shoreline.  The cost function is given below.  Can you see how it was developed?  Finish the problem.

2.  Find the point on the graph of the parabola whose equation is y = x2 + 3 closest to the point (2,2).  State your answer accurate to 4 digits to the right of the decimal.  Click on the picture at the right to see an animation.  The parabola is graphed in blue and the function giving the distance from the parabola to the point (2,2) is graphed in red.  The green line segments represent the changing distance between the parabola and the point. EXTRA CREDIT:  Click here to see a blow-up of the animation relating to the picture at the right with the addition of the line (drawn in blue) tangent to the graph of the parabola at the blue animated point.  The EC is to tell me what I am trying to show you by drawing this tangent.  If you can figure this out then you can receive more EC by proving it.

3.  Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart.  They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post.  Where should the stake be placed to use the least wire.  Click here to see an animation of the possibilities along with the total length of wire function.

Alternate Quiz Problem

A farmer has 200 meters of fence to use in fencing in a rectangular pasture bordering on a straight river.  The farmer is not going to fence along the river so he will only be putting fence along the three sides shown in green in the picture on the right.  Find the function that gives the total enclosed area as a function of x and find the dimensions of the rectangular pasture so fenced that will yield the maximum enclosed area.  Click here or on the picture to see an animation.  In the animation the graph in red represents the total area divided by 50 (in order to fit on the screen) as a function of x .

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats