 #1
JTHMZeroOne
 2
 0
Alright so currently I'm learning about using zeros of first and second derivites to find relative max and mins in an equation. That stuff is pretty simply. Also we're learning to differentiate equations in relation to a variable that is not in the problem. But, I cannot figure out these problems and would appreciate help.
1. A cylindrical can of radius r inches and height h inches is to contain 36 cubic inches of liquid. The volume of the can is V= (pi)r^2h cubic inches and the total surface area is S = area of the sides + area of top and bottom = 2(pi)rh + 2(pi)r^2 square inches. The material for the top and the bottom of the can costs 22 cents per square inch and the material for the sides costs 18 cents per square inch. Find the values of r and h that minimize the cost of constructing the can.
I started by differentiating V=(pi)r^2h in respect to dh/dsa and dr/sa I came up with dr/dsa = (r/2h)*(dh/dsa). I don't really know where to go from here though.
2. A Natural gas pipe is to be laid from the main gas line to the island airport located on the shoreline of an island as shown in the figure. the island is 2.2 miles from the mainland and the main gas line is 8 miles away from a point that is directly landward of the island. The cost of the pipe is 2.5 times times as great in the water as on land. At what point should the pipe meet the mainland shore in order to minimize total costs?
island

2.2 miles

Land8 milesmain line
I used the pythagereon therom to determine the length of the third side (8.2970) but I don't know how to create a diffirential equation relating price in so I can find the minimum using it's derivitive.
3. The number n, in thousands, of a certain type of radial tire that is sold each month and the monthly expenses a, in thousands of dollars, for advertising that tire are related by the equation .15a9n+an+.02an^28.13=0. A tire is manufcaturer currently selling 3.5 thousand tires each month and spending $6000 per month on advertising. If plans call for increasing the advertising expenses by .4 thousand per month, how quickly should sales increase.
sorry, I don't know where to begin on this problem.
Any help would be greatly appreciated. Thanks for your time.
Homework Statement
1. A cylindrical can of radius r inches and height h inches is to contain 36 cubic inches of liquid. The volume of the can is V= (pi)r^2h cubic inches and the total surface area is S = area of the sides + area of top and bottom = 2(pi)rh + 2(pi)r^2 square inches. The material for the top and the bottom of the can costs 22 cents per square inch and the material for the sides costs 18 cents per square inch. Find the values of r and h that minimize the cost of constructing the can.
I started by differentiating V=(pi)r^2h in respect to dh/dsa and dr/sa I came up with dr/dsa = (r/2h)*(dh/dsa). I don't really know where to go from here though.
2. A Natural gas pipe is to be laid from the main gas line to the island airport located on the shoreline of an island as shown in the figure. the island is 2.2 miles from the mainland and the main gas line is 8 miles away from a point that is directly landward of the island. The cost of the pipe is 2.5 times times as great in the water as on land. At what point should the pipe meet the mainland shore in order to minimize total costs?
island

2.2 miles

Land8 milesmain line
I used the pythagereon therom to determine the length of the third side (8.2970) but I don't know how to create a diffirential equation relating price in so I can find the minimum using it's derivitive.
3. The number n, in thousands, of a certain type of radial tire that is sold each month and the monthly expenses a, in thousands of dollars, for advertising that tire are related by the equation .15a9n+an+.02an^28.13=0. A tire is manufcaturer currently selling 3.5 thousand tires each month and spending $6000 per month on advertising. If plans call for increasing the advertising expenses by .4 thousand per month, how quickly should sales increase.
sorry, I don't know where to begin on this problem.
Any help would be greatly appreciated. Thanks for your time.