MATH-2450 HANDOUT 6 (Lesson 15 – Applied Maximum and Minimum Problems)
1.Find two positive real numbers whose sum is 40 and whose product is a maximum.
2.A box with a square base and open top is to have a volume of 4 ft 3, find the dimensions that require the least amount of material.
3.A fence 8 feet tall stands on level ground and runs parallel to a tall building. If the fence is 1 foot from the building, then find the shortest ladder that will extend from the ground over the fence to the wall of the building.
4.Find the dimensions of the rectangle of maximum area that can be inscribed in an equilateral triangle of side of a, if two vertices of the rectangle lie on one side of the triangle.
5.Find the point on the graph of that is the closest to point (3, 1).
6.Find the point on the lower half of the ellipse that is nearest the point (-1, 0).
7.An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting out a square from each corner and then bending up the sides. Find the size of the corner square which will produce a box having the largest possible volume and then the volume.
8.A page of a book is to have an area of 90 square inches, with 1-inch margins at the bottom and sides and a -inch margin at the top. Find the dimensions of the page which will allow the largest printed area.
9.A manufacturer sells a certain article to dealers at a rate of $20 each if 50 or less are ordered. If more than 50 are ordered (up to 600) the price per article is reduced at a rate of 2 cents times the number ordered. What size order will produce the maximum amount of money for the manufacturer?
10.A veterinarian has 100 feet of fencing and wishes to construct six dog kennels by first building a fence around a rectangular region and then subdividing that region into six smaller rectangles by placing five fences parallel to one of the sides. What dimensions of the region will maximize the total area?
11.Find the dimensions of the right circular cone of maximum volume that can be inscribed in a sphere of radius a.
12.Find the dimensions of the rectangle of maximum area that can be inscribed in a semicircle of radius 4 feet, if two vertices of the rectangle lie on the diameter of the semicircle.
13.A closed box is to be constructed having a volume of 96 ft 3. The length of the bottom (and hence, the top) of the box is four times the width of the bottom (and hence, the top). Find the dimensions that require the least amount of material.
14.Steve wishes to fence a rectangular region of area 500 square yards. What are the dimensions which require the least amount of fencing?
15.An open rectangular box is to be constructed having a volume of 288 in 3 and a square base. The material for the bottom of the box costs 8 cents per square inch and the material for the four sides costs 5 cents per square inch. Find the dimensions of the box which is the cheapest to construct.
16.Bill can only afford to buy 100 yards of fencing. If he uses the fencing to enclose his rectangular garden, then find the largest area that he can enclose.
17.A long rectangular sheet of metal, 16 inches wide, is made into a rain gutter by turning up two sides at right angles to the sheet. What are the dimensions of the gutter with the largest cross sectional area?
18.A rectangle is inscribed in an isosceles triangle whose base is 48 feet and sides are 30 feet. If two of the vertices of the rectangle lie on the base of the triangle, then find the dimensions of the largest rectangle that can be inscribed in the triangle.
19.A farmer wishes to fence a rectangular pasture with a total area of 1000 yd 2, and he wants to divide it into two parts with a fence across the middle. Fencing around the outside costs $5.00 per yard, but he can use less expensive fencing at $2.00 per yard as a divider. Find the dimensions of the region which will cost the minimum to construct.
20.A company has 100 yards of fencing. It wants to use the fencing to enclose a rectangular region next to a warehouse. If they do not fence in the side next to the warehouse, what is the largest area that can be enclosed?