8  Lab 8: Optimization

Objective

The objective of this lab is to provide students with hands-on experience in solving optimization problems using calculus. Through various real-world examples, students will learn to identify and formulate optimization problems, apply constraints, and use derivatives to find optimal solutions.

Specific Goals

  1. Identify Objectives and Constraints:
    • Understand the problem statement to identify what needs to be maximized or minimized (objective).
    • Recognize and incorporate any given constraints into the optimization problem.
  2. Formulate Optimization Problems:
    • Translate real-world problems into mathematical models.
    • Express the objective function in terms of a single variable using constraints.
  3. Apply Derivative Techniques:
    • Use first and second derivative tests to find critical points and determine their nature (local maxima or minima).
    • Verify and interpret solutions in the context of the original problem.

Problems to Solve

  1. Maximum Area of a Rectangle:
    • Determine the dimensions of a rectangle with a perimeter of 10 that maximizes the area.
  2. Minimum Perimeter:
    • Find the dimensions of a rectangle with an area of 100 that minimizes the perimeter.
  3. Constraint by an Equation:
    • Find positive numbers \(x\) and \(y\) satisfying \(xy = 12\) such that the sum \(2x + y\) is minimized.
  4. Biggest Rectangle Inside a Semi-Circle:
    • Determine the dimensions of a rectangle constructed within a semicircle of radius 5 that maximizes the area.
  5. Shortest Ladder:
    • Find the length of the shortest ladder that clears an 8-foot fence and reaches a house 3 feet away.

By completing these exercises, students will gain practical experience in formulating and solving optimization problems, enhancing their problem-solving skills and understanding of calculus applications.

8.1 Maximum Area of a Rectangle

Of all rectangles with a perimeter of 10, which one has the maximum area?
(Give the dimensions)

First we need to identify our objective, in this case is to maximize the area of a rectangle:

\[\boxed{max(A=xy)}\]

Second we need to identify our constraint, in this case the perimeter of our rectangle is equal to 10:

\[x+y+x+y = 10 \] \[ \boxed{x+y =5} \]

Now we need to include our constraint in our objective

\[\boxed{constraint \rightarrow objective}\]

then the area function become dependent in a single variable

\[\boxed{A(x,y)\rightarrow A(x)}\]

Now with this I can work, and find what is the maximum, applying what we learned in previous sections, i.e.

\[\begin{align*} y &= 5-x\\ \Rightarrow\\ A(x,y)=xy \rightarrow A(x)&=x(5-x)\\ &=5x-x^2\\ \Rightarrow\\ A'&=5-2x\\ A'=0 \Leftrightarrow x&=\frac{5}{2}\\ A''&=-2 \qquad(\text{concavity down}) \end{align*}\]

The for \(x=\frac{5}{2}\) we have an absolute maximum
(only extremum in a concavity down region).

Then \(y= 5-x = 5-\frac{5}{2} = \frac{5}{2}\)

Therefore (\(x=\frac{5}{2},y=\frac{5}{2}\)) are the dimensions which maximize the area of a rectangle(square) with a constraint of 10 to be the perimeter.

Using maxima-cas:

(%i1) define(A(x),x*(5-x));

\[\mathtt{(\textit{%o}_{1})}\quad A\left(x\right):=\left(5-x\right)\,x\]

(%i2) define(dAdx(x),diff(A(x),x,1));

\[\mathtt{(\textit{%o}_{2})}\quad \textit{dAdx}\left(x\right):=5-2\,x\]

(%i3) define(ddAdx(x),diff(A(x),x,2));

\[\mathtt{(\textit{%o}_{3})}\quad \textit{ddAdx}\left(x\right):=-2\]

(%i4) solns: solve(dAdx(x)=0,x);

\[\mathtt{(\textit{%o}_{4})}\quad \left[ x=\frac{5}{2} \right] \]

(%i5) ddAdx(rhs(solns[1]));

\[\mathtt{(\textit{%o}_{5})}\quad -2\]

(%i6) plot2d(A(x),[x,0,5]);

8.2 Minimum Perimeter

Of all rectangles of area 100, which one has the minimum perimeter?
(Give the dimensions)

8.3 Constraint by an Equation

Find the positive numbers \(x\) and \(y\) satisfying the equation: \[xy = 12\] such that the sum: \[2x + y\] is as small as possible.

8.4 Biggest Rectangle inside of a semi-circle.

A rectangle is constructed with its base on the diameter of a semicircle with radius 5 and its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?

8.5 Shortest Ladder

An 8-foot-tall fence runs parallel to the side of a house 3 feet away. What is the length of the shortest ladder that clears the fence and reaches the house? Assume the vertical wall of the house and the horizontal ground have infinite extent.

8.6 For each of the previous sections:

  • take pictures of your handwritten work, submit them to the dropbox.
  • Link each picture to the corresponding section of your wxmaxima document1 (Cell -> Insert Image).
  • be sure your code run correctly by pressing Shift+Enter in each code chunk.
  • be sure the whole document display correctly, by running Ctrl+R.
  • Submit on the corresponding Dropbox folder, with the corresponding name (FINAL_Lab8_Calc1_YOURLASTNAME_mmddyy.wxmx).

  1. pictures need to be in the same folder of your .wxmx file↩︎