Objectives
- Solve optimization problems
Section 3.7 Optimization
Subsection 3.7.1 Before Class
https://mymedia.ou.edu/media/3.7-1/1_dby87rvfhttps://mymedia.ou.edu/media/3.7-2/1_4wf321rgSubsubsection 3.7.1.1 Optimization
Example 3.7.1.
Consider the function \(f(x) = -x^2 + 6x + 11\text{.}\)- Use the methods of Section 3.3 to find the critical point of \(f(x)\text{;}\) give your answer as an ordered pair.
- Sketch the graph of \(f(x)\) to visually confirm your answer from (a).
- Now suppose that \(f(x)\) gives the profit (in hundred dollars) that a company makes from a certain product, when \(x\) thousand units are sold. The critical point you found in (a) has a physical meaning now; what is it?
Solution.
- \(\displaystyle (3,20)\)
![The graph of \(f(x) = -x^2+6x+11\) on the interval \([-3,3]\text{.}\)](generated/latex-image/image-128.svg)
- It gives the location of maximum profit.
Example 3.7.2.
A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field with the largest area?
Solution.
Let \(x\) be the length of the side parallel to the river, and \(y\) be length of the perpendicular sides. Then, \(x = 1200\text{ ft}\) and \(y = 600 \text{ ft}\)
Strategy for Optimization.
- Draw a picture of the situation (if possible). Label all pieces of your diagram with a variable that will not confuse you later.
- Use the picture (or the description) to find equation(s) which relate to the problem.
- Identify the equation which will be optimized. Use the equations from (2) and algebra to reduce the variables in the optimizing equation, and simplify.
- Take the derivative of your optimizing equation, set it equal to zero, and solve.
- If necessary, use your answer to find the remaining pieces of information.
The key to optimization problems is practice. Many optimization problems follow a similar pattern, and require only slight modification from other problems. Practice is the only way that one will learn and recognize these patterns and tricks.
Example 3.7.3.
A cylindrical can is to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can.
Solution.
Let \(r\) be the radius of the can, \(h\) be the height. Then, \(r = \sqrt[3]{\dfrac{500}{\pi}}\) and \(h = 2\sqrt[3]{\dfrac{500}{\pi}}\)
Example 3.7.4.
Find the point on the parabola \(y^2 = 2x\) that is closest to the point \((1,4)\text{.}\)
Solution.
\((2,2)\)
Subsection 3.7.2 Pre-Class Activities
Example 3.7.5.
Find two numbers whose difference is 150 and whose product is a minimum.
Solution.
Let \(x\) and \(y\) be the numbers. Then, \(x = 75\) and \(y = -75\text{.}\)
Example 3.7.6.
The sum of two positive numbers is 20. What is the smallest possible value of the sum of their squares?
Solution.
200
Example 3.7.7.
What is the maximum vertical distance between the line \(y=x+2\) and the parabola \(y=x^2\) for \(-1\leq x\leq 2\text{?}\)
Solution.
\(\dfrac{9}{4}\)
Example 3.7.8.
How are you feeling about optimization problems so far? What can we do to make you feel better about them?
Solution.
Answers vary
Subsection 3.7.3 In Class
Example 3.7.9.
A woman launches her boat from point \(A\) on a bank of a straight river, 3 miles wide, and wants to reach point \(B\text{,}\) 8 miles downstream on the opposite bank, as quickly as possible. She could: row her boat directly across the river to point \(C\) and run to point \(B\text{;}\) row directly to \(B\text{;}\) or, row to some intermediate point \(D\) and run to \(B\text{.}\) She can can row 5 mi/h and run 6 mi/h; where should she land in order to reach \(B\) as soon as possible?
Solution.
Let \(x\) be the distance between \(C\) and \(D\text{.}\) Then, \(x = \dfrac{15}{\sqrt{11}} \text{ mi}\)
Example 3.7.10.
Find the area of the largest rectangle which can be inscribed in a semicircle of radius \(r\text{.}\)
Solution.
\(A = r^2\)
Example 3.7.11.
A box with a square base and open top must have a volume of 32,000 cm\(^3\text{.}\) Find the dimensions of the box that minimize the amount of material used.
Solution.
Let \(x\) be the length of the base, and \(y\) be the height. Then, \(x = 40\text{ cm}\) and \(y = 20 \text{ cm}\text{.}\)
Example 3.7.12.
A rectangular storage container with an open top is to have a volume of 10 m\(^3\text{.}\) The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.
Solution.
The minimal cost is \(20\lrpar{\sqrt[3]{\dfrac{9}{2}}}^2 + \dfrac{180}{\sqrt[3]{9/2}}\) dollars
Example 3.7.13.
Find the area of the largest rectangle that can be inscribed in the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} = 1\text{.}\)
Solution.
\(A = 2ab\)
Example 3.7.14.
A right circular cylinder is inscribed in a sphere of radius \(r\text{.}\) Find the largest possible volume of such a cylinder.
Solution.
\(V = \dfrac{4\pi}{3\sqrt{3}}r^3\)
Example 3.7.15.
A poster is to have an area of 180 in\(^2\) with 1-inch margins at the bottom and sides, and a 2-inch margin at the top. What dimensions will give the largest printed area?
Solution.
Let \(x\) be the printed length and \(y\) be the printed height. Then, \(x = \dfrac{180}{\sqrt{270}}-2\) in and \(y = \sqrt{270} - 3\) in.
Example 3.7.16.
If the two equal sides of an isosceles triangle have length \(a\text{,}\) find the length of the third side that maximizes the area of the triangle.
Solution.
The side length is \(\dfrac{2a}{\sqrt{3}}\) units
Example 3.7.17.
A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle \(\theta\text{.}\) How should \(\theta\) be chosen so that the gutter will carry the maximum amount of water?
Solution.
\(\theta = \dfrac{\pi}{3}\)
Subsection 3.7.4 After Class Activities
Example 3.7.18.
A farmer with 1000 feet of fencing wants to enclose a rectangular area, and divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?
Solution.
Let \(x\) from the length of the side perpendicular to the pen fencing, and \(y\) be the length of the side parallel to the pen fencing. Then, \(x = 250\) feet and \(y = 100\) feet.
Example 3.7.19.
Find the point on the curve \(y = \sqrt{x}\) that is closest to the point \((4,0)\text{.}\)
Solution.
\(\lrpar{\dfrac{7}{2},\sqrt{\dfrac{7}{2}}}\)
Example 3.7.20.
A piece of wire 10 inches long is cut into two pieces. One piece is bent into a square, and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) maximized? (b) minimized?
Solution.
For the minimal area, cut \(\dfrac{\sqrt{3}/18}{1/8 + \sqrt{3}/18}\) inches from either end; for the maximal area, don’t cut the wire.
Example 3.7.21.
Find an equation of the line through the point \((5,3)\) that cuts off the least area from quadrant one.
Solution.
\(y = -\dfrac{3}{5}x + 6\)
