Transcribed Text
a. A rectangular pen is built with one side against a barn. If 800 m of fencing are used for the other three sides of the pen, what dimensions maximize the area of the pen?
b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 m2 (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used?
Barn
100
100
100
100
a Let A be the area of the rectangular pen and let x be the length of the sides perpendicular to the barn Write the objective function in a form that does not include the length of the side parallel to the barn.
A=
(Type an expression.)
The interval of interest of the objective function is
(Simplify your answer Type your answer in interval notation Do not use commas in the individual endpoints. )
To maximize the area of the pen, the sides perpendicular to the barn should be
m long and the side parallel to the barn should be
(Type exact answers, using radicals as needed.)
m long
b. Let x be the length of the sides perpendicular to the barn and let L be the total length of fence needed Write the objective function
L=
(Type an expression.)
The interval of interest of the objective function is
(Simplify your answer Type your answer in interval notation Do not use commas in the individual endpoints. )
To minimize the amount of fence that must be used, each of the sides perpendicular to the barn should be
(Type exact answers, using radicals as needed.)
m long and each of the sides parallel to the barn should be
m long.
Use the figures to calculate the left and right Riemann sums for f on the given
y
Q
Ay
interval and the given value of n.
3
f(x) = +1
x
4
3
4
f(x)===1
f(x) =x = + 1 on [1,5]; n   4
3
Q
[
x
x
1
2
3
4
5
0 1 2 3 4 5
The left Riemann sum for f is
(Round to two decimal places as needed.)
The right Riemann sum for fis

(Round to two decimal places as needed.)
These solutions may offer stepbystep problemsolving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skillbuilding and practice.
Unethical use is strictly forbidden.
a) A= 800x  2x²
The interval of interest of the objective of function is [0,400]...