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What number exceeds its square by the maximum amount?

What number exceeds its square by the maximum amount?

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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1. What number exceeds its square by the maximum amount? 2. What positive number added to its reciprocal gives the minimum sum? 3. The sum of two numbers is k. Find the minimum value of the sun of their squares. 4. The sum of two numbers is k. Find the minimum value of the sun of their cubes. 5. What should be the shape of a rectangular field of a given area, if it is to be enclosed by the least amount of fencing? 6. A rectangular field of 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? 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 minimum. 8. 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 foot, along the sides $1 per foot, find the dimensions of the largest lot which can be thus fenced in for $300. 9. The perimeter of an isosceles triangle is P inches. Find the maximum area. 10.A lot has the form of a right triangle, with perpendicular sides 60 and 80 feet long. Find the length and width of the largest rectangular building that can be erected, facing the hypotenuse of the triangle. 11. From a strip of tin 14 inches a trapezoidal gutter is to be made by bending up the sides at an angle of 45. Find the width of the base for greatest carrying capacity.
Transcribed Image Text:1. What number exceeds its square by the maximum amount? 2. What positive number added to its reciprocal gives the minimum sum? 3. The sum of two numbers is k. Find the minimum value of the sun of their squares. 4. The sum of two numbers is k. Find the minimum value of the sun of their cubes. 5. What should be the shape of a rectangular field of a given area, if it is to be enclosed by the least amount of fencing? 6. A rectangular field of 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? 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 minimum. 8. 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 foot, along the sides $1 per foot, find the dimensions of the largest lot which can be thus fenced in for $300. 9. The perimeter of an isosceles triangle is P inches. Find the maximum area. 10.A lot has the form of a right triangle, with perpendicular sides 60 and 80 feet long. Find the length and width of the largest rectangular building that can be erected, facing the hypotenuse of the triangle. 11. From a strip of tin 14 inches a trapezoidal gutter is to be made by bending up the sides at an angle of 45. Find the width of the base for greatest carrying capacity.
12.A ship lies 6 miles from shore, and opposite a point 10 miles farther along the shore another ship lies 18 miles offshore. A boat from the first ship is to land a passenger and then proceed to the other ship. What is the least distance the boat can travel? 13. Two posts, one 8 feet high and the other 12 feet high, stand 15 ft apart. They are to be supported by wires attached to a single stake at ground level. The wires running to the tops of the posts. Where should the stake be placed, to use the least amount of wire? 14. Find the point on the curve d2xy = 3 that is nearest the point (4a, 0). 15. Find the shortest distance from the point (5, 0) to the curve 2y² =x². 16. Find the shortest distance from the point (0, 8a) to the curve ax² = y³. 17.A silo consists of a cylinder surmounted by a hemisphere. If the floor, walls, and roof are equally expensive per unit area, find the most economical proportion. 18. An oil can consists of a cylinder surmounted by a cone. If the diameter of the cone is five-sixths of its height, find the most economical proportions. 19. Find the largest right pyramid with a square base that can be inscribed in a sphere of radius a. 20.A man on an island 12 miles south of a straight beach wishes to reach a point on shore 20 miles east. If a motorboat, making 20 miles per hour, can be hired at the rate of $2.00 per hour for the time it is actually used, and the cost of land transportation is $0.06 per mile, how much must he pay for the trip?
Transcribed Image Text:12.A ship lies 6 miles from shore, and opposite a point 10 miles farther along the shore another ship lies 18 miles offshore. A boat from the first ship is to land a passenger and then proceed to the other ship. What is the least distance the boat can travel? 13. Two posts, one 8 feet high and the other 12 feet high, stand 15 ft apart. They are to be supported by wires attached to a single stake at ground level. The wires running to the tops of the posts. Where should the stake be placed, to use the least amount of wire? 14. Find the point on the curve d2xy = 3 that is nearest the point (4a, 0). 15. Find the shortest distance from the point (5, 0) to the curve 2y² =x². 16. Find the shortest distance from the point (0, 8a) to the curve ax² = y³. 17.A silo consists of a cylinder surmounted by a hemisphere. If the floor, walls, and roof are equally expensive per unit area, find the most economical proportion. 18. An oil can consists of a cylinder surmounted by a cone. If the diameter of the cone is five-sixths of its height, find the most economical proportions. 19. Find the largest right pyramid with a square base that can be inscribed in a sphere of radius a. 20.A man on an island 12 miles south of a straight beach wishes to reach a point on shore 20 miles east. If a motorboat, making 20 miles per hour, can be hired at the rate of $2.00 per hour for the time it is actually used, and the cost of land transportation is $0.06 per mile, how much must he pay for the trip?
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