Posted by Rachel | Posted in Coffee and Accessories | Posted on 09-01-2009
Tags: blog, bunn coffee filters 12 cup, coffee, coffee filters 12, coffee filters 12 cup, coffee filters 12 cup coffeemaker, news, technology, toddy coffee filters 12 pack, web2.0
Calculus Optimization Problems?
1) A drilling rig 12 miles offshore is to be connected by a pipe to a refinery onshore, 20 miles down the coast from the rig. If underwater pipe costs $50,000/mile and land-based pipe costs $30,000/mile, what values of x and y give the least expensive connection?
2) Coffee is draining from a conical filter into a cylindrival coffeepot at the rate of 10 in^3/min. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep? How fast is the level in the cone falling then?
1) Draw a sketch of the layout. Create a triangle where the line from the rig to the shore is 12 miles long and is at right angles to the shore. Draw a line from the intersection of the first line with the shore to the refinery. That line is 20 miles long. Pick some arbitrary point on the line, and divide the line into 2 segments, one being x units long and the other being 20 – x units long. Join that point to the rig to form the hypotenuse of the triangle. Then:
Cost C = 50 X √(144 + x²) + 30(20 – x)
dC/dx = 50 / (x/(√144 + x²) – 30 = 0
Solve for x
x = 9,
The point at which the line should intersect the shore is 9 miles along the shore. The length of the line on shore is therefore 11. The length of the line under water is √(144 + x²) = 15 miles
The least expensive solution is 15 X 50,000$ + 9 X $30,000
$1,020,000
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