Maxima and Minima
Problem 1
What number exceeds its square by the
maximum amount?
Solution 1
Problem 4
The sum of two numbers is k. Find the
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Problem 2 minimum value of the sum of their cubes.
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What positive number added to its reciprocal Solution 4
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gives the minimum sum?
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Solution 2
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blem 3
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The sum of two numbers is k. Find the
minimum value of the sum of their squares.
Solution 3
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Number Problems
Problem 5
The sum of two positive numbers is 2. Find
the smallest value possible for the sum of the
cube of one number and the square of the
other.
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Problem 7
Find two numbers whose sum is a, if the
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product of one by the cube of the other is to
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Solution 7
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Problem 6
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Find two numbers whose sum is a, if the
product of one to the square of the other is to
be a minimum.
Solution 6 Problem 8
Find two numbers whose sum is a, if the
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product of the square of one by the cube of Problem 10
the other is to be a maximum. 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?
Solution 10
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Problem 11
rs e A rectangular lot is to be fenced off along a
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highway. If the fence on the highway costs m
dollars per yard, on the other sides n dollars
per yard, find the area of the largest lot that
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can be fenced off for k dollars.
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Solution 11
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Rectangular Lot Problems
Problem 9
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What should be the shape of a rectangular
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field of a given area, if it is to be enclosed by
the least amount of fencing?
Solution 9
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Problem 14
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
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fence along the front costs $1.50 per foot,
along the sides $1 per foot, find the
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dimensions of the largest lot which can be
thus fenced in for $300.
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Problem 12 Solution 14
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A rectangular field of fixed area is to be
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enclosed and divided into three lots by
parallels to one of the sides. What should be
the relative dimensions of the field to make
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the amount of fencing minimum?
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Solution 12
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Problem 13
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Do Ex. 12 with the words "three lots" replaced
by "five lots".
Solution 13
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Box open at the top
Problem 15
A box is to be made of a piece of cardboard 9
inches square by cutting equal squares out of
the corners and turning up the sides. Find the
volume of the largest box that can be made in
this way.
Solution:
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Problem 16
Find the volume of the largest box that can be
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made by cutting equal squares out of the
corners of a piece of cardboard of dimensions
15 inches by 24 inches, and then turning up
the sides.
Solution:
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SOLVED PROBLEMS IN MAXIMA AND
MINIMA
Problem 21
Find the rectangle of maximum perimeter
inscribed in a given circle.
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Problem 24
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Problem 22 Solve Problem 23 if the box has an open top.
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If the hypotenuse of the right triangle is given,
show that the area is maximum when the
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triangle is isosceles.
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Solution:
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Problem 25
Find the most economical proportions of a
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Problem 23 quart can.
Find the most economical proportions for a Solution:
covered box of fixed volume whose base is a
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rectangle with one side three times as long as
the other.
Solution:
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Problem 27
Find the most economical proportions for a
box with an open top and a square base.
Solution:
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Problem 28
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The perimeter of an isosceles triangle is P
Problem 26
rs e inches. Find the maximum area.
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Find the most economical proportions for a
cylindrical cup.
Solution:
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Problem 29
The sum of the length and girth of a container
of square cross section is a inches. Find the
maximum volume.
Solution:
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Problem 30
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Find the proportion of the circular cylinder of
largest volume that can be inscribed in a
given sphere.
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Problem 33
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.
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Problem 32
Find the dimension of the largest rectangular
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building that can be placed on a right-
triangular lot, facing one of the perpendicular
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sides.
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