A rectangular box with open top and has a square base is a school project of two senior high school students. However, they are wondering what would be the minimum dimensions of their box if it will have a fixed volume of 62.5 cm³. Steps 1. Draw a diagram. List what is asked on the problem and label the diagram with relevant data. 2. Write the constraint and the optimization equations. 3. Substitute the constraint equations, to the corresponding length, width and height of the optimization equation. 4. Simplify and take its first derivative. 5. Set the equation to zero and solve for the x value (critical point). 6. If there are two critical values, we have to check which one will give a sensible answer by substituting them to the volume equation. 7. Test the x value. Substitute it to the second derivative and check whether the answer is less than or greater than zero. 8. Substitute the maximum x value to the simplified constraint equation to solve for I and w. 9. Solve for the dimensions. Solution Constraint equation: Optimization equation: height= length width=
A rectangular box with open top and has a square base is a school project of two senior high school students. However, they are wondering what would be the minimum dimensions of their box if it will have a fixed volume of 62.5 cm³. Steps 1. Draw a diagram. List what is asked on the problem and label the diagram with relevant data. 2. Write the constraint and the optimization equations. 3. Substitute the constraint equations, to the corresponding length, width and height of the optimization equation. 4. Simplify and take its first derivative. 5. Set the equation to zero and solve for the x value (critical point). 6. If there are two critical values, we have to check which one will give a sensible answer by substituting them to the volume equation. 7. Test the x value. Substitute it to the second derivative and check whether the answer is less than or greater than zero. 8. Substitute the maximum x value to the simplified constraint equation to solve for I and w. 9. Solve for the dimensions. Solution Constraint equation: Optimization equation: height= length width=
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Answer the Problems, Show your Complete Solutions (This is all about Calculus: Optimization Problems).
I already provided the answers I just need the COMPLETE SOLUTIONS.
a. Length = width = 5 m, Height = 2.5 m
b. x = 135 m, y = 67.5 m
Transcribed Image Text:a.A rectangular box with open top and has a square base is a school project of two
senior high school students. However, they are wondering what would be the
minimum dimensions of their box if it will have a fixed volume of 62.5 cm³.
Solution
Steps
1. Draw a diagram. List what is asked
on the problem and label the
diagram with relevant data.
Constraint equation:
2. Write the constraint
and the
optimization equations.
Optimization equation:
3. Substitute the constraint equations,
to the corresponding
length, width and height of the
optimization equation.
4. Simplify and take its first derivative.
5. Set the equation to zero and solve for
the x value (critical point).
6. If there are two critical values, we
have to check which one will give a
sensible answer by substituting
them to the volume equation.
7. Test the x value. Substitute it to the
second derivative and check
whether the answer is less than or
greater than zero.
8. Substitute the maximum x value to
the simplified constraint equation to
solve for l and w.
height =
length =
width =
%3D
9. Solve for the dimensions.
Transcribed Image Text:P.A resort owner wants to enclose a beachfront area for swimming activities. Based on
her plan, only 3 sides will be fenced with 270-meter rope and floats, while the
shoreline part will be open. Determine the dimensions of the 3 sides of the rectangle
that will give a maximum area.
Solution
Steps
1. Draw a diagram. List what is asked
on the problem and label the
diagram with relevant data.
Constraint equation:
2. Write
optimization equations.
the constraint and
the
Optimization equation:
3. On the constraint equation, solve
for y. Then, substitute that equation
to its corresponding y on the
optimization equation.
4. Simplify and take its first derivative.
5. Set the equation to zero and solve for
the x value (critical point).
6. Test the x value. Substitute it to the
second derivative and check whether
the answer is less than or greater
than zero.
7. Substitute the minimum x value to
the simplified constraint equation to | dimensions of the rectangle.
solve for y.
x = _
and y =
the
are
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