Application:

Maxima and Minima

1. What positive number added to its reciprocal gives the minimum sum?
2. The sum of two numbers is k. Find the minimum value of the sum of their squares.
3. The sum of two positive numbers is 4. Find the smallest value possible for the sum of the cube of one number and the square of the other.
4. Find two numbers whose sum is a., if the product of one by the square of the others is to be a maximum.
5. Find two numbers whose sum is a., if the product of one by the cube of the others is to be a maximum.
6. Find two numbers whose sum is a., if the product of square of one by the cube of the others is to be a maximum.
7. What should be the shape of a rectangular field of given area, if it is to be enclosed by the least amount of fencing?
8. A rectangular field of a given area is to be fenced off along the bank of a river. If no fence is needed along the river, what is the shape of the
rectangle requiring the least amount of fencing?
9. A rectangular lot is to be fenced off along a highway. If the fence of the highway cost 𝑚 dollars per yard; on the other sides n dollars per yard,
Find the area of the largest lot that can be fenced off for k dollars.
10. A rectangular field of fixed area is to be enclosed and divided into three lots by parallels to one of the sides. What should be the relative
dimensions of the field to make the amount of fencing a minimum.
11. A rectangular field of fixed area is to be enclosed and divided into five lots by parallels to one of the sides. What should be the relative dimensions
of the field to make the amount of fencing a minimum.
12. A rectangular lot is bounded at the back by a river. No fence is needed along the river and there is to be 24 ft opening in front. If the fence along
the front costs $1.50 per feet, along the sides $1 per feet, Find the dimension ,of the largest lot which can be thus fenced in for $300.
13. A box is to be made of a piece of cardboard 9 in. square by cutting equal squares out of the corner and turning up the sides. Find the volume
of the largest box that can be made in this way.
14. Find the volume of the largest box that can be made by cutting equal squares out of the corner of a piece of cardboard. of dimension 15in x 24in
and then turning up the sides.
15. Find the rectangle of the maximum perimeter inscribed in a given circle.
16. An open field is bounded by a lake with a straight shoreline. A rectangular enclosure is to be constructed using 500 ft of fencing along three
sides and the lake as a natural boundary on the fourth side. What dimensions will maximize the enclosed area? What is the maximum area?
17. Ryan has 800 ft of fencing. He wishes to form a rectangular enclosure and then divide it into three sections by running two lengths of fence
parallel to one side. What should the dimensions of the enclosure be in order to maximize the enclosed area?
18. 20 meters of fencing are to be laid out in the shape of a right triangle. What should its dimensions be in order to maximize the enclosed area?
19. A piece of wire 100 inches long is to be used to form a square and/or a circle. Determine their (a) maximum and (b) minimum combined area.
20. Find the maximum area of a rectangle inscribed in a semicircle of radius 5 inches if its base lies along the diameter of the semicircle.
21. An open box is to be constructed from a 12- × 12-inch piece of cardboard by cutting away squares of equal size from the four corners and folding
up the sides. Determine the size of the cutout that maximizes the volume of the box.
22. A window is to be constructed in the shape of an equilateral triangle on top of a rectangle. If its perimeter is to be 600 cm, what is the maximum
possible area of the window?
23. Postal regulations require that the sum of the length and girth of a rectangular package may not exceed 108 inches (the girth is the perimeter
of an end of the box). What is the maximum volume of a package with square ends that meets this criteria?
24. A rectangle is inscribed in a right triangle whose sides are 5, 12, and 13 inches. Two adjacent sides of the rectangle lie along the legs of the
triangle. What are the dimensions of the rectangle of maximum area? What is the maximum area?
25. Find the dimensions of the right circular cylinder of maximum volume that can be inscribed in a right circular cone whose radius is 3 in and
whose height is 10 in. What is the maximum volume?
26. What is the minimum amount of fencing needed to construct a rectangular enclosure containing 1800 ft 2 using a river as a natural boundary on
one side?
27. An open rectangular box is to have a base twice as long as it is wide. If its volume must be 972 cm 3, what dimensions will minimize the amount
of material used in its construction?
28. Find the points on the parabola y = x 2 closest to the point (0, 1).
29. A publisher wants to print a book whose pages are each to have an area of 96 in 2. The margins are to be 1 in on each of three sides and 2 in on
the fourth side to allow room for binding. What dimensions will allow the maximum area for the printed region?
30. A closed cylindrical can must have a volume of 1000 in 3. What dimensions will minimize its surface area?
31. A closed cylindrical can must have a volume of 1000 in 3. Its lateral surface is to be constructed from a rectangular piece of metal and its top and
bottom are to be stamped from square pieces of metal and the rest of the square discarded. What dimensions will minimize the amount of
metal needed in the construction of the can?
𝑥2 𝑦2
32. A rectangle is to be inscribed in the ellipse + = 1. Determine its maximum possible area.
200 50
Related Rates
33. Air is being pumped into a spherical balloon at a rate of 5 cm3 /min. Determine the rate at which the radius of the balloon is increasing when
the diameter of the balloon is 20 cm.
34. A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of
14 ft/sec. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?
35. A tank of water in the shape of a cone is leaking water at a constant rate of 2 ft 3 / hour. The base radius of the tank is 5 ft and the height of the
tank is 14 ft. At what rate is the depth of the water in the tank changing when the depth of the water is 6 ft?
36. A tank of water in the shape of a cone is leaking water at a constant rate of 2 ft 3 / hour. The base radius of the tank is 5 ft and the height of the
tank is 14 ft. At what rate is the depth of the water in the tank changing when the depth of the water is 6 ft? At what rate is the radius of the
top of the water in the tank changing when the depth of the water is 6 ft?
37. A light is on the top of a 12 ft tall pole and a 5ft 6in tall person is walking away from the pole at a rate of 2 ft/sec. At what rate is the tip of the
shadow moving away from the pole when the person is 25 ft from the pole?
38. A light is on the top of a 12 ft tall pole and a 5ft 6in tall person is walking away from the pole at a rate of 2 ft/sec. At what rate is the tip of the
shadow moving away from the person when the person is 25 ft from the pole?
39. A trough of water is 8 meters deep and its ends are in the shape of isosceles triangles whose width is 5 meters and height is 2 meters. If water
is being pumped in at a constant rate of 6m3/sec. At what rate is the height of the water changing when the water has a height of 120 cm?
40. Two legs of a right triangle are each 70 cm. If one leg grows at the rate of 5 cm/min and the other shrinks at the rate of 5 cm/min, (a) How fast
is the hypotenuse of the triangle changing 2 minutes later? (b) How fast is the area of the triangle changing 2 minutes later?
41. A fisherman has a fish at the end of his line, which is being reeled in at the rate of 2 ft/sec from a bridge 30 ft above the water. At what speed
is the fish moving through the water toward the bridge when the amount of line out is 50 ft? (Assume the fish is at the surface of the water and
there is no sag in the line.)
42. Sand is being dumped from a dump truck at the rate of 10 ft 3/min and forms a pile in the shape of a cone whose height is always half its radius.
How fast is its height rising when the pile is 5 ft high?
43. A radar station is 2000 ft from the launch site of a rocket. If the rocket is launched vertically at the rate of 500 ft/sec, how fast is the distance
between the radar station and the rocket changing10 seconds later?
44. A baseball diamond is a square whose sides are 90 ft long. If a batter hits a ball and runs to first base at the rate of 20 ft/sec, how fast is his
distance from second base changing when he has run 50 ft?