A rancher wants to fence in an area of 2000000 square feet in a rectangular field and then
divide it in half with a fence down the middle parallel to one side. What is the shortest
length of fence that the rancher can use?

A fence is to be built to enclose a rectangular area of 240 square
feet. The fence along three sides is to be made of material that
costs 4 dollars per foot, and the material for the fourth side
costs 13 dollars per foot. Find the length L and width W
(with W &le L) of the enclosure that is most economical to
construct.

A hippogriff rancher wants to enclose a rectangular area and then divide it into
four pens with fencing parallel to one side of the rectangle (see the figure below).
He has 600 feet of fencing available to complete the job. What is the largest
possible total area of the four pens?

A box is to be made out of a 6 cm by 20 cm piece of cardboard. Squares
of side length x cm will be cut out of each corner, and then the ends
and sides will be folded up to form a box with an open top.

(a) Express the volume V of the box as a function of x.

V =

(b) Give the domain of V in interval notation. (Use the fact that length and volume must be positive.)

(c) Find the length L, width W, and height H of the resulting
box that maximizes the volume. (Assume that W &le L).

If 1300 square centimeters of material is available to make a box with
a square base and an open top, What is the largest possible volume of the box that can be found?
Volume = cubic centimeters.

A rectangular storage container with an open top is to have a volume of 50 cubic meters.
The length of its base is twice the width. Material for the base costs 50 dollars per
square meter. Material for the sides costs 7 dollars per square meter. Find the cost
of materials for the cheapest such container.
Minimum cost = .

Find the dimensions of the rectangle of largest area that can be
inscribed in an equilateral triangle with sides of length 10 if
one side of the rectangle lies on the base of the triangle.

A rectangle is inscribed with its base on the x-axis and its
upper corners on the parabola
. What are the dimensions of such a rectangle with the greatest
possible area?

A Norman window has the shape of a semicircle atop a rectangle so that the diameter
of the semicircle is equal to the width of the rectangle. What is the area of the largest
possible Norman window with a perimeter of 20 feet?

A cylinder is inscribed in a right circular cone of height 3
and radius (at the base) equal to 3. What are the dimensions of such a cylinder
which has maximum volume?

A fence 3 feet tall runs parallel to a tall building at a distance of 6 feet from
the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.

Here are some hints for finding a solution:
Use the angle that the ladder makes with the ground to define the
position of the ladder and draw a picture of the ladder leaning against the wall of the building and just touching the top of the fence.

If the ladder makes an angle 0.85 radians with the ground, touches the top
of the fence and just reaches the wall, calculate the distance along
the ladder from the ground to the top of the fence.

The distance along the ladder from the top of the fence to the wall
is

Using these hints write a function L(x) which gives the total length of a ladder which touches
the ground at an angle x, touches the top of the fence and
just reaches the wall.
L(x) = .
Use this function to find the length of the shortest ladder which will
clear the fence.
The length of the shortest ladder is feet.

Centerville is the headquarters of Greedy Cablevision Inc. The
cable company is about to expand service to two nearby towns,
Springfield and Shelbyville. There needs to be cable connecting
Centerville to both towns. The idea is to save on the cost of
cable by arranging the cable in a Y-shaped configuation.
Centerville is located at
(9,0) in the xy-plane, Springfield is at (0,6), and
Shelbyville is at (0,- 6). The cable runs from Centerville
to some point (x,0) on the x-axis where it splits into two branches going to
Springfield and Shelbyville. Find the location (x,0)
that will minimize the amount of cable between the 3 towns and
compute the amount of cable needed. Justify your answer.

To solve this problem we need to minimize the following function
of x:
f(x)=
We find that f(x) has a critical number at x=
To verify that f(x) has a minimum at this critical number we compute
the second derivative f''(x) and find that its value at the critical
number is , a positive number.
Thus the minimum length of cable needed is

A small town is situated on an island in the pacific ocean that lies exactly 6 miles
from P, which is the nearest point to the island along a perfectly
straight shoreline. 10
miles down the shoreline from P is the closest source of fresh
water. If it costs 1.7 times as much money to lay pipe in the
water as it does on land, how far down the shoreline from P
should the pipe from the island reach land in order to minimize
the total construction costs?