does anyone understand these.
1. Two roads intersect at right angles. A water spring is located 60 m from one road and 50 m from the other road. A straight path is to be laid out to pass the spring from one road to the other. Find the least area that can be bounded by the roads and the path.
2.A commercial packing company wishes to build a shipping box. The box is to be made from a rectangular piece of cardboard 10 m long and 5 m wide by cutting equal squares from each corner and then folding up the remaining sides. What is the maximum volume that the box can have?
3. A rectangular piece of metal is 33.0 cm long and 26.4 cm wide. A rectangular box is to be formed by cutting a square out of each corner and turning up the remaining sides. The box does not have a top. Determine the size of the square that should be cut out so that the box has a maximum volume.
4.A pig farmer wishes to fence his rectangular field so that it will contain an area of 167 square meters. If the cost is 2 dollars per meter along two sides and one end and 4 dollars per meter along the other end, calculate the dimensions which give the least cost for the fence.
5. A farmer plans to build a storage bin. This bin is to be an open-top rectangular tank with square base and is to have a volume of 12 cubic meters. The cost of the bottom is $17 per square meter and the cost of the sides $2 per square meter. Find the most economical dimensions for the tank.
6.The voltage in a circuit is given by the formula V = 13t3 - 57t. Find the values for t for which V is a maximum or a minimum.
7. The power delivered to a circuit by a 15 volt generator of internal resistance 2 ohms when the current is i amperes, is 15i - 2i2 watts. For what current will the generator provide maximum power?
In each case you have a function that needs to be minimized. The functions seem to be pretty easy to determine - area of a triangle, perimeter of a square, etc. In later problems, they are actually given.
To find the minimum or maximum, set the derivative to zero and solve.
I don't think it happens on these problems, but the minimum or maximum can also occur when the derivative doesn't exist or at the endpoints. So you need to check all three cases.
In fact i would differ a bit, in that we cannot consider a maxima or minima at the end points. it is because of the definition of these points. For example local maxima is the value of the function which is maximum in the neighborhood of that point etc.