Optimization

Section 3.3 Optimization

Optimization problems in calculus are always asking to maximize or minimize some quantity. Typical phrases that indicate an optimization problem include: "Find the largest ..." or "Find the minimum ..."

To solve an optimization problem you need to do the following:

Find a function of one variable that models the situation described in the given problem.
Use the Calculus tools to find the critical numbers and determine whether they correspond to a local maximum or minimum.

Solve the following optimization problems:

Find the absolute maximum and minimum values of $f(x)=3x^2-9x$ on the interval $[-1,2]\text{.}$
Find the absolute maximum and minimum values of $f(x)=x^3-12x-5$ on the interval $[-4,6]\text{.}$ Clearly explain your reasoning.
If $a$ and $b$ are positive numbers, find the maximum value of $\ds f(x)=x^a(1-x)^b\text{.}$
Find all critical points of the function $f(x)=|3x-5|$ on the interval $[-3,2]\text{.}$ Also find all maxima and minima of this function on $[-3,2]\text{,}$ both local and global.
The sum of two positive numbers is 12. What is the smallest possible value of the sum of their squares? Show your reasoning.
If $a$ and $b$ are positive numbers, find the $x$ coordinate which gives the absolute maximum value of $f(x)=x^a(1-x)^b$ on the interval $[0,1]\text{.}$
Find the point on the curve $x+y^2=0$ that is closest to the point $(0,-3)\text{.}$
A straight piece of wire 40 cm long is cut into two pieces. One piece is bent into a circle and the other is bent into a square. How should wire be cut so that the total area of both the circle and square is minimized?
A straight piece of wire 28 cm long is cut into two pieces. One piece is bent into a square (i.e. dimensions $x$ times $x\text{.}$) The other piece is bent into a rectangle with aspect ratio three (i.e. dimensions $y$ times $3y\text{.}$) What are the dimensions, in centimetres, of the square and the rectangle such that the sum of their areas is minimized?
With a straight piece of wire 4m long, you are to create an equilateral triangle and a square, or either one only. Suppose a piece of wire of length $x$ metres is bent into a triangle and the remainder is bent into a square. Find the value of $x$ which maximizes the total area of both the triangle and the square.
Show that a perimeter of $2\sqrt{5}R$ is the largest possible perimeter of a rectangle inscribed in a semicircle of radius $R\text{,}$ with one side of the rectangle lying along the diameter of the semicircle. See Figure 3.3.1.

Figure 3.3.1. A semicircle and a rectangle
A rectangle with sides parallel to the coordinate axes is to be inscribed in the region enclosed by the graphs of $y=x^2$ and $y=4$ so that its perimeter has maximum length.
1. Sketch the region under consideration.
2. Supposing that the $x$-coordinate of the bottom right vertex of the rectangle is $a\text{,}$ find a formula which expresses $P\text{,}$ the length of the perimeter, in terms of $a\text{.}$
3. Find the value of $a$ which gives the maximum value of $P\text{,}$ and explain why you know that this value of $a$ gives a maximum.
4. What is the maximum value of $P\text{,}$ the length of the perimeter of the rectangle?
Find the dimensions of the rectangle of largest area that has its base on the $x$-axis and its other two vertices above the $x$-axis and lying on the parabola $y=12-x^2\text{.}$
A farmer has 400 feet of fencing with which to build a rectangular pen. He will use part of an existing straight wall 100 feet long as part of one side of the perimeter of the pen. What is the maximum area that can be enclosed?
A $10\sqrt{2}$ ft wall stands 5 ft from a building, see Figure 3.3.2. Find the length $L$ of the shortest ladder, supported by the wall, that reaches from the ground to the building.

Figure 3.3.2. A building, a wall, and a ladder
An attacking player (Gretzky) is skating with the puck along the boards as shown. As Gretzky proceeds, the apparent angle $\alpha$ between the opponent's goal posts first increases, then decreases.

Figure 3.3.3. Goal!
1. Using dimensions given in Figure 3.3.3, find an expression for $\alpha$ in terms of the distance $x$ from Gretzky to the goal line.
2. Assume that Gretzky's chance of scoring is greatest when $\alpha$ is maximum (this may be the case if the opposing team has “pulled” their goalie). At which distance $x$ from the goal line should Gretzky shoot the puck? It is clear that $\alpha$ is very small when $x=35$ and $x=0\text{,}$ so there is no need to check the endpoints of the domain $[0,35]\text{.}$
In an elliptical sport field we want to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of $\ds \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\text{.}$ Find the length $2x$ and width $2y$ of the pitch (in terms of $a$ and $b$) that maximize the area of the pitch. [Hint: express the area of the pitch as a function of $x$ only.]
The top and bottom margins of a poster are each 6 cm, and the side margins are each 4 cm. If the area of the printed material on the poster (that is, the area between the margins) is fixed at 384 cm$^2\text{,}$ find the dimensions of the poster with the smallest total area.
A poster is to have an area of 180 in$^2$ with 1-inch margins at the bottom and the sides and a 2 inch margin at the top. What dimensions will give the largest printed area? See Figure 3.3.4.

Figure 3.3.4. A poster
Each rectangular page of a book must contain 30 cm$^2$ of printed text, and each page must have 2 cm margins at top and bottom, and a 1 cm margin at each side. What is the minimum possible area of such a page?
Maya is 2 km offshore in a boat and wishes to reach a coastal village which is 6 km down a straight shoreline from the point on the shore nearest to the boat. She can row at 2 km/hr and run at 5 km/hr. Where should she land her boat to reach the village in the least amount of time?
A hiker is in the woods at a distance of 3 km from a straight road. She would like to reach a supply store which is located 10 km down the road from the nearest point on the road to her. Assuming that she hikes through the woods at a rate of 2 km/hr and walks along the road at a rate of 4 km/h, what point on the road should she hike through the woods to so as to minimize her travel time? See Fig 3.3.5.

Figure 3.3.5. A hiker, a road, and a store
A rectangular box has a square base with edge length $x$ of at least 1 unit. The total surface area of its six sides is 150 square units.
1. Express the volume $V$ of this box as a function of $x\text{.}$
2. Find the domain of $V(x)\text{.}$
3. Find the dimensions of the box in part (a) with the greatest possible volume. What is this greatest possible volume?
An open-top box is to have a square base and a volume of 10 m$^3\text{.}$ The cost per square metre of material is $\$ 5$ for the bottom and $\$ 2$ for the four sides. Let $x$ and $y$ be lengths of the box's width and height respectively. Let $C$ be the total cost of material required to make the box.
1. Express $C$ as a function of $x$ and find its domain.
2. Find the dimensions of the box so that the cost of materials is minimized. What is this minimum cost?
An open-top box is to have a square base and a volume of 13500 cm$^3\text{.}$ Find the dimensions of the box that minimize the amount of material used.
A water trough is to be made from a long strip of tin 6 ft wide by bending up at the angle $\theta$ a 2 ft strip at each side. What angle $\theta$ would maximize the cross sectional area, and thus the volume, of the trough? (See Figure 3.3.6.)

Figure 3.3.6. A water trough
Find the dimensions of the right circular cylinder with greatest volume that can be inscribed in a right circular cone of radius 8 cm and height 12 cm. See Figure 3.3.7.

Figure 3.3.7. A cylinder and a cone
Find the dimension of the right circular cylinder of maximum volume that can be inscribed in a right circular cone of radius $R$ and height $H\text{.}$
A hollow plastic cylinder with a circular base and open top is to be made and 10 m$^2$ of plastic is available. Find the dimensions of the cylinder that give the maximum volume and find the value of the maximum volume.
An open-topped cylindrical pot is to have volume 250 cm$^3\text{.}$The material for the bottom of the pot costs 4 cents per cm$^2\text{;}$ that for its curved side costs 2 cents per cm$^2\text{.}$ What dimensions will minimize the total cost of this pot?
A cylindrical can without a top is made to contain 1,000 cm$^2$ of liquid. Find the dimensions that will minimize the cost of the material to make the can.
Cylindrical soup cans are to be manufactured to contain a given volume $V\text{.}$ No waste is involved in cutting the material for the vertical side of each can, but each top and bottom which are circles of radius $r\text{,}$ are cut from a square that measures $2r$ units on each side. Thus the material used to manufacture each soup can has an area of $A=2\pi rh+8r^2$ square units.
1. How much material is wasted in making each soup can?
2. Find the ratio of the height to diameter for the most economical can (i.e. requiring the least amount of material for manufacture.)
3. Use either the first or second derivative test to verify that you have minimized the amount of material used for making each can.
A storage container is to be made in the form of a right circular cylinder and have a volume of $28\pi$ m$^3\text{.}$ Material for the top of the container costs $\$ 5$ per square metre and material for the side and base costs $\$ 2$ per square metre. What dimensions will minimize the total cost of the container?
Show that the volume of the largest cone that can be inscribed inside a sphere of radius $R$ is $\ds \frac{32\pi R^3}{81}\text{.}$
The sound level measured in watts per square metre, varies in direct proportion to the power of the source and inversely as the square of the distance from the source, so that is given by $\ds y=kPx^{-2}\text{,}$ where $y$ is the sound level, $P$ is the source power, $x$ is the distance form the source, and $k$ is a positive constant. Two beach parties, 100 metres apart, are playing loud music on their portable stereos. The second party's stereo has 64 times as much power as the first. The music approximates the white noise, so the power from the two sources arriving at a point between them adds, without any concern about whether the sources are in or out of phase. To what point on the line segment between the two parties should I go, if I wish to enjoy as much quiet as possible? Demonstrate that you have found an absolute minimum, not just a relative minimum.