Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible.
A farmer with 2000 feet of fencing wants to enclose a rectangular area that borders a straight river, and then divide it into four pens with fencing perpendicular to the river. Find the dimensions of the rectangle that yield the largest possible total area of the four pens.
You want to construct a fence around a rectangular region of area 1225 sq ft. Find the dimensions of the rectangular region to minimize the amount of material needed.
A poster is to have an area of 216 square inches with 1 inch margins at the bottom and sides and a 2 inch margin at the top. Find the exact dimensions that will give the largest printed area.
A box with an open top is to be constructed from a square piece of cardboard, 6 feet wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.
If 1200 square inches of material is available to make a box with a square base and no top, find the dimensions that maximize the volume of the box.
A box with a square base, an open top, and a volume of 10,976 cubic inches is to be constructed. Find the dimensions of the box that will minimize the amount of material used.
Find the point on the line \(3x - y - 1 = 0\) that is closest to the point \((4, 1)\).
Find the points on the ellipse \(x^2 + 4y^2 = 4\) that are farthest away from the point \((0, 1)\).
A farmer has $1500 available to build a fence along a straight river to create two identical rectangular pastures. The materials for the side parallel to the river cost $6 per foot and the materials for the three sides perpendicular to the river cost $5 per foot. Find the dimensions for which the total area of the pastures will be as large as possible, assuming no fence is needed along the river.