See also the
Browse High School Calculus
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Maximizing the volume of a box.
Maximizing the volume of a cylinder.
Volume of a tank.
What is a derivative?
- The Logistic Model for Population Growth [03/29/1998]
I'm trying to solve the logistic model of population growth for P(t).
- Longest Ladder [11/30/2001]
Two hallways, one 8 ft. wide and other 4 ft. wide, meet to form a right
angle. What is the longest ladder that can go around the corner where the
- Manipulating Limits [10/13/2003]
How do I use the limit definition to find the derivative at a point
where the function becomes undefined?
- Math Questions [3/19/1995]
What is the volume of the solid of revolution created by the function, y
= cos(x) and y = (x+1)/3, when these are revolved about the x-axis?
- Maximize Area of Trapezoid [07/28/2003]
Given an isosceles trapezoid with three of its sides of length 10 cm,
find the fourth side so that the area is maximized.
- Maximizing Area [05/02/1997]
A wire is cut into two pieces. One piece is shaped into a square, the
other into an equilateral triangle. How should the wire be cut to
maximize the area enclosed by the the two pieces?
- Maximizing Oranges [06/29/2002]
An orchard has 800 orange trees, each of which yields 120 oranges per
season. For each new tree that is added to the orchard, the output
per individual tree decreases by 2 oranges per season. Determine the
number of trees that would maximize the orchard's output.
- Maximizing Revenues [07/29/2002]
For every increase of $1 in ticket price, 200 fewer people will attend
each game. What ticket price will maximize revenues?
- Maximizing the Volume of a Box [11/05/1996]
What dimensions should a rectangular piece of paper have to maximize the
volume of the box made by cutting the corners out and folding?
- Maximizing the Volume of a Box: Find Size of Square Cutout [07/09/1999]
If I have a rectangular card of size AxB, how can I find the size of the
square cutout that maximizes the volume of the box produced when the
edges are folded up?
- Maximizing the Volume of a Cylinder [12/3/1995]
Find the dimensions of the cylinder of maximum volume that can be
inscribed in a cone having a diameter of 40 cm and a height of 30 cm.
Show that the maximum area of the cylinder is 4/9 the volume of the cone.
- Maximizing the Volume of a Cylinder [7/24/1996]
I need to maximize the volume of a right-circular cylinder that fits
inside a sphere of radius 1 m.
- Maximizing the Volume of a Rain Gutter [10/21/2003]
A rain gutter is to be constructed from a metal sheet of width 30cm,
by bending up one-third of the sheet on each side by an angle (theta)
from the horizontal (theta = zero represents the unbent sheet).
Determine what theta should be chosen so that the gutter will carry
the most water when it is full.
- Maximizing Window Area [02/24/1997]
Maximize the area of a Norman window (rectangular with a semicircle on
top) while minimizing the length of the perimeter.
- Maximum Area for Irrigation Channel [02/08/2002]
An irrigation channel made of concrete is to have a cross section in the
form of an isosceles trapezoid, three of whose sides are 4 feet long. How
should it be shaped to have a maximum area?
- Maximum Area of an Arc [12/4/1995]
A string of length 25cm is to be formed into the arc of a circle so that
the area of the segment formed will be a maximum.
- Maximum Area of Inscribed Triangle [12/10/2001]
An isosceles triangle is inscribed in a circle of radius R. Find the
value of Theta that maximizes the area of the triangle.
- Maximum Points and the Second Derivative [04/24/1998]
When the second derivative of a point is a negative, is the point said to
be a maximum?
- Maximum Surface Area for Total Edge Length [07/14/2002]
A piece of wire of total length L units is used to form the nine edges
of a prism whose ends are equilateral triangles and whose other faces
are rectangles. What is the maximum surface area of this prism?
- Maximum Volume: Making a Box from a Sheet of Paper [10/21/1999]
I am making a box out of a sheet of paper by cutting squares out of the
corners. How can I show that the largest possible volume occurs when the
side of the square is 1/6 the length of the sheet of paper?
- Maximum Volume of a Box [04/13/1997]
A rectangular sheet of cardboard measures 16cm by 6cm. Equal squares are
cut out of each corner and the sides are turned up to form an open
rectangular box. What is the maximum volume of the box?
- Max/Min Applications of Derivatives [05/31/1998]
I need help with these max/min applications of derivatives. If 40
passengers hire a train...
- Max/Min Problems with 3-D Shapes [03/23/2009]
I am trying to find the dimensions of a cylindrical can that will use
the minimal surface area for a given volume. I've done maximum and
minimum problems with two-dimensional shapes. Does the process work
the same way with three-dimensional objects?
- Meaning of Derivative [10/12/1996]
What's a plain English meaning of the derivative?
- The Meaning of 'dx' in an Integral [02/22/2002]
What meaning is attached to the 'dx' at the end of an integral?
- Mean Value Theorem [04/18/1999]
Which number satisfies the conclusion of the Mean Value Theorem for f(x)
= sin (x/2)?
- Melting a Cube [12/20/2001]
How long does it take for an ice cube to melt?
- Methods of Approximation [7/31/1996]
Is there an alternative to Newton's method of approximation to solve
y^y = 25?
- Method to Integrate sqrt(1-x^2) [9/11/1996]
What is the integral of [sqrt(1-x^2)]dx from [-1,1]?
- Minimizing an Enclosed Area [06/02/1999]
How can I determine the position of a point P on the line y = 2 such that
the line segments OP and PQ and arc OQ of the curve x = y^2 enclose
minimum total area?
- Minimizing the Cost of a Box [11/14/1995]
A closed box with a square base is to contain 252 cubic feet. The bottom
costs $5 per square foot, the top costs $2 per square foot, and the sides
cost $3 per square foot. Find the dimensions that will minimize the cost.
- Minimizing the Cost of Phone Line Construction [12/3/1995]
A telephone company has to run a line from point A on one side of a river
to another point B that is on the other side, 5km down from the point
opposite A. The river is uniformly 12km wide. The company can run the
line along the shoreline to a point C and then under the river to B. The
cost of the line along the shore is $1000 per km and the cost under the
river is twice as much. Where should point C be to minimize the cost?
- Minimizing the Length of a Crease [11/11/1999]
How can I minimize the the length (L) of the crease if one of the corners
of an 8.5" x 11" sheet of paper is lifted up and placed on the opposite
longer edge, and then the paper is smoothed flat?
- Minimizing the Sums of Squares [06/12/1998]
Find two numbers such that their sum is 20, and the sum of their squares
is as small as possible.
- Minimizing the Surface Area of a Can [05/22/2000]
What coke can dimensions would use the least amount of aluminum while
still holding 375 ml?
- Minimizing the Time to Rescue [11/29/1999]
A man is drowning in the water 300 meters down the beach and 100 meters
out from where I'm standing. I can run 5 meters/second and swim 3
- Minimum and Maximum Pollution [04/26/2000]
Find the points of minimum and maximum pollution between two plants
releasing 60 and 240 ppm of pollution...
- Minimum, Maximum Value of a Two-variable Function [6/27/1996]
Find f min and f max where x and y are real numbers and f(x,y) = 2(sin
x)(cos y)+3(sin x)(siny)+6(cos x).
- Minimum Value Problem [7/14/1995]
I need to find the minimum value of E for E = be^(-ar) - dr^(-6) where b,
a, and d are constants and r is the variable.
- Min, Max of 3-Variable Function [8/16/1996]
Find all the critical points and determine their nature for the function
z = x^3 - 6xy + y^3.