Folded boxes (a). Squares with sides of length x are cut

A house is located at each corner of a square with side lengths of 1 mi. What is the length of the shortest road system with straight roads that connects all of the houses by roads (that is, a road system that allows one to drive from any house to any other house)? (Hint: Place two points inside the square at which roads meet.) (Source: Halmos, Problems for Mathematicians Young and Old.)

A rancher is building a horse pen on the corner of her property using 1000 ft of fencing. Because of the unusual shape of her property, the pen must be built in the shape of a trapezoid (see figure). a. Determine the lengths of the sides that maximize the area of the pen. b. Suppose there is already a fence along the side of the property opposite the side of length y. Find the lengths of the sides that maximize the area of the pen, using 1000 ft of fencing.

A window consists of a rectangular pane of clear glass surmounted by a semicircular pane of tinted glass. The clear glass transmits twice as much light per unit of surface area as the tinted glass. Of all such windows with a fixed perimeter P, what are the dimensions of the window that transmits the most light?

Suppose you are standing in a field near a straight section of railroad tracks just as the locomotive of a train passes the point nearest to you, which is \(\frac{1}{4}\) mi away. The train, with length \(\frac{1}{3}\) mi, is traveling at 20 mi/hr. If you start running in a straight line across the field, how slowly can you run and still catch the train? In which direction should you run?

An arbelos is the region enclosed by three mutually tangent semicircles; it is the region inside the larger semicircle and outside the two smaller semicircles (see figure). a. Given an arbelos in which the diameter of the largest circle is 1, what positions of point B maximize the area of the arbelos? b. Show that the area of the arbelos is the area of a circle whose diameter is the distance BD in the figure.

A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball. Assume point C is the closest point on the edge of the beach from the tennis ball (see figure). a. Assume the dog runs at r m/s and swims at s m/s, where r > s. Also assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball. b. Verify that the value of y that minimizes the time it takes to retrieve the ball is \(y=\frac{x}{\sqrt{r / s+1} \sqrt{r / s-1}}\). c. If the dog runs at 8 m/s and swims at 1 m/s, what ratio y/x produces the fastest retrieving time? d. A dog named Elvis who runs at 6.4 m/s and swims at 0.910 m/s was found to use an average ratio y/x of 0.144 to retrieve his ball. Does Elvis appear to know calculus? (Source: College Mathematics Journal, May 2003)

a. Two poles of heights m and n are separated by a horizontal distance d. A rope is stretched from the top of one pole to the ground and then to the top of the other pole. Show that the configuration that requires the least amount of rope occurs when \(\theta_{1}=\theta_{2}\) (see figure). b. Fermat's Principle states that when light travels between two points in the same medium (at a constant speed), it travels on the path that minimizes the travel time. Show that when light from a source A reflects off of a surface and is received at point B, the angle of incidence equals the angle of reflection, or \(\theta_{1}=\theta_{2}\) (see figure).

A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume that gasoline costs $p/gallon and the vehicle gets g miles per gallon. Also, assume that the driver earns $w/hour. a. A plausible function to describe how gas mileage (in mi/gal) varies with speed is g(v) = v(85 – v)/60. Evaluate g(0), 8(40), and g(60) and explain why these values are reasonable. b. At what speed does the gas mileage function have its maximum? c. Explain why the cost of a trip of length L miles is C(v) = Lp/g(v) + Lw/v. d. Let L = 400 mi, p = $4/gal, and w = $20/hr. At what (constant) speed should the vehicle be driven to minimize the cost of the trip? e. Should the optimal speed be increased or decreased compared with part (d)] if L is increased from 400 mi to 500 mi? Explain. f. Should the optimal speed be increased or decreased compared with part (d)] if p is increased from $4/gal to $4.20/gal? Explain. g. Should the optimal speed be increased or decreased (compared with part (d)] if w is decreased from $20/hr to $15/hr? Explain.

Suppose that a light source at A is in a medium in which light travels at a speed \(v_{1}\), and the point B is in a medium in which light travels at a speed \(v_{2}\) (see figure). Using Fermat's Principle, which states that light travels along the path that requires the minimum travel time (Exercise 59), show that the path taken between points A and B satisfies \(\left(\sin \theta_{1}\right) / v_{1}=\left(\sin \theta_{2}\right) / v_{2}\).

A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half planes is \(\theta\). Prove that for a given tree and fixed angle \(\theta\), the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.

Many species of small mammals (such as flying squirrels and marsupial gliders) have the ability to walk and glide. Recent research suggests that these animals choose the most energy-efficient means of travel. According to one empirical model, the energy required for a glider with body mass m to walk a horizontal distance D is \(8.46 \mathrm{Dm}^{2 / 3}\) (where m is measured in grams, D is measured in meters, and energy is measured in microliters of oxygen consumed in respiration). The energy cost of climbing to a height D tan \(\theta\) and gliding at an angle of \(\theta\) (below the horizontal, with \(\theta=0\) representing perfectly horizontal flight and \(\theta>45^{\circ}\) representing controlled falling) a horizontal distance D is modeled by 1.36 mD tan \(\theta\). Therefore, the function \(S(m, \theta)=8.46 m^{2 / 3}-1.36 m \tan \theta\) gives the energy difference per horizontal meter traveled between walking and gliding: If S > O for given values of m and \(\theta\), then it is more costly to walk than glide. a. For what glide angles is it more efficient for a 200 gram animal to glide rather than walk? b. Find the threshold function \(\theta=g(m)\) that gives the curve along which walking and gliding are equally efficient. Is it an increasing or decreasing function of body mass? c. In order to make gliding more efficient than walking, do larger gliders have a larger or smaller selection of glide angles that they can use? d. Let \(\theta=25^{\circ}\) (a typical glide angle) and graph S as a function of m for \(0 \leq m \leq 3000\) (in grams). For what values of misgliding more efficient? e. For \(\theta=25^{\circ}\), what value of m (call it \(m^{*}\)) maximizes S? f. Does \(m^{*}\), as defined in part (e), increase or decrease with increasing \(\theta\)? That is, as a glider reduces its glide angle, does its optimal size become larger or smaller? g. Assuming Dumbo is a gliding elephant whose weight is one metric ton (\(10^{6}\) g), what glide angle would Dumbo use to be more efficient at gliding than walking? (Source: Energetic savings and the body size distribution of gliding mammals, Roman Dial, Evolutionary Ecology Research, 5 (2003): 1151-1162.)

Fill in the blanks: The goal of an optimization problem is to find the maximum or minimum value of the __________ function subject to the __________.

If the objective function involves more than one independent variable, how are the extra variables eliminated?

If the objective function is \(Q=x^{2} y\) and you know that x + y = 10, write the objective function first in terms of x and then in terms of y.

Suppose you wish to minimize a continuous objective function on a closed interval, but you find that it has only a single local maximum. Where should you look for the solution to the problem?

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

Minimum perimeter rectangles Of all rectangles with a fixed area A, which one has the minimum perimeter? (Give the dimensions in terms of A.)

Maximum product What two nonnegative real numbers with a sum of 23 have the largest possible product?

Minimum sum What two positive real numbers whose product is 50 have the smallest possible sum?

Maximum length What two nonnegative real numbers a and b whose sum is 23 maximize \(a^{2}+b^{2}\)? Minimize \(a^{2}+b^{2}\)?

Pen problems a. A rectangular pen is built with one side against a barn. Two hundred meters 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 rectanguIar pens against a barn, each with an area of \(100\mathrm{\ m}^2\) (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used? c. Two rectangular pens are built against a barn. Two hundred meters of fencing are to be used for the three sides and the diagonal dividing fence (see figure). What dimensions maximize the area of the pen?

Minimum surface area box Of all boxes with a square base and a volume of \(100\mathrm{\ m}^3\), which one has the minimum surface area? (Give its dimensions.)

Maximum volume box Suppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 108 in. What are the dimensions and volume of a square based box with the greatest volume under these conditions?

A square-based, box-shaped shipping crate is designed to have a volume of \(16 \mathrm{ft}^{3}\). The material used to make the base costs twice as much (per \(\mathrm{ft}^{2}\)) as the material in the sides, and the material used to make the top costs half as much (per \(\mathrm{ft}^{2}\)) as the material in the sides. What are the dimensions of the crate that minimize the cost of materials?

A man wishes to get from an initial point on the shore of a circular pond with radius 1 mi to a point on the shore directly opposite (on the other end of the diameter). He plans to swim from the initial point to another point on the shore and then walk along the shore to the terminal point. a. If he swims at 2 mi/hr and walks at 4 mi/hr, what are the minimum and maximum times for the trip? b. If he swims at 2 mi/hr and walks at 1.5 mi/hr, what are the minimum and maximum times for the trip? c. If he swims at 2 mi/hr, what is the minimum walking speed for which it is quickest to walk the entire distance?

A 10-ft-tall fence runs parallel to the wall of a house at a distance of 4 ft. Find the length of the shortest ladder that extends from the ground, over the fence, to the house. Assume the vertical wall of the house and the horizontal ground have infinite extent.

A boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 6 mi from a restaurant on the shore. A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. a. If she walks at 3 mi/hr and rows at 2 mi/hr, at which point on the shore should she land to minimize the total travel time? b. If she walks at 3 mi/hr, what is the minimum speed at which she must row so that the quickest way to the restaurant is to row directly with no walking)?

An 8-ft-tall fence runs parallel to the wall of a house at a distance of 5 ft. Find the length of the shortest ladder that extends from the ground, over the fence, to the house. Assume that the vertical wall of the house is 20 ft high and the horizontal ground extends 20 ft from the fence.

A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola \(y=16-x^{2}\). What are the dimensions of the rectangle with the maximum area? What is the area?

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

A piece of wire 60 cm in length is cut, and the resulting two pieces are formed to make a circle and a square. Where should the wire be cut to (a) minimize and (b) maximize the combined area of the circle and the square?

A cone is constructed by cutting a sector of angle \(\theta\) from a circular sheet of metal with radius 20 cm. The cut sheet is then folded up and welded (see figure). What angle \(\theta\) maximizes the volume of the cone?

Imagine a flat-bottomed cylindrical pot with a circular cross section of radius 4 in. A marble with radius 0 < r <4 in is placed in the bottom of the pot. What is the radius of the marble that requires the most water to cover it completely?

a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area? b. Is it possible to construct a rectangle with a greater area than that found in part (a) by placing one side of the rectangle on the line y = 10 - 2x, and the vertices not on that line on the positive x- and y-axes? Find the dimensions of the rectangle of maximum area that can be constructed in this way.

A rectangular flower garden with an area of \(30 \mathrm{~m}^{2}\) is surrounded by a grass border 1 m wide on two sides and 2 m wide on the other two sides (see figure). What dimensions of the garden minimize the combined area of the garden and borders?

Several mathematical stories originated with the second wedding of the mathematician and astronomer Johannes Kepler. Here is one: While shopping for wine for his wedding, Kepler noticed that the price of a barrel of wine (here assumed to be a cylinder) was determined solely by the length d of a dipstick that was inserted diagonally through a hole in the top of the barrel to the edge of the base of the barrel (see figure). Kepler realized that this measurement does not determine the volume of the barrel and that for a fixed value of d, the volume varies with the radius r and height h of the barrel. For a fixed value of d, what is the ratio r/h that maximizes the volume of the barrel?

a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. b. Suppose that in part (a) the original piece of cardboard is a square with sides of length \(\ell\). Find the volume of the largest box that can be formed in this way. c. Suppose that in part (a) the original piece of cardboard is a rectangle with sides of length \(\ell\) and L. Holding \(\ell\) fixed, find the size of the corner squares x that maximizes the volume of the box as \(L \rightarrow \infty\). (Source: Mathematics Teacher, November 2002)

A grain silo consists of a cylindrical concrete tower surmounted by a metal hemispherical dome. The metal in the dome costs 1.5 times as much as the concrete (per unit of surface area). If the volume of the silo is \(750 \mathrm{~m}^{3}\), what are the dimensions of the silo (radius and height of the cylindrical tower) that minimize the cost of the materials? Assume the silo has no floor and no flat ceiling under the dome.

a. Classical problem Find the radius and height of a cylindrical soda can with a volume of \(354 \quad \mathrm{cm}^{3}\) that minimize the surface area. b. Real problem Compare your answer in part (a) to a real soda can, which has a volume of \(354 \quad \mathrm{cm}^{3}\), a radius of 3.1 cm, and a height of 12.0 cm, to conclude that real soda cans do not seem to have an optimal design. Then use the fact that real soda cans have a double thickness in their top and bottom surfaces to find the radius and height that minimizes the surface area of a real can (the surface areas of the top and bottom are now twice their values in part (a)). Are these dimensions closer to the dimensions of a real soda can?

Right circular cones of height h and radius r are attached to each end of a right circular cylinder of height h and radius r, forming a double-pointed object. For a given surface area A, what are the dimensions r and h that maximize the volume of the object?

An auditorium with a flat floor has a large screen on one wall. The lower edge of the screen is 3 ft above eye level and the upper edge of the screen is 10 ft above eye level (see figure). How far from the screen should you stand to maximize your viewing angle?

A searchlight is 100 m from the nearest point on a straight highway (see figure). As it rotates, the searchlight casts a horizontal beam that intersects the highway in a point. If the light revolves at a rate of \(\pi / 6\) rad/s, find the rate at which the beam sweeps along the highway as a function of \(\theta\). For what value of \(\theta\) is this rate minimized?

An observer stands 20 m from the bottom of a Ferris wheel on a line that is perpendicular to the face of the wheel, with her eyes at the level of the bottom of the wheel. The wheel revolves at a rate of \(\pi\) rad/min and the observer's line of sight with a specific seat on the Ferris wheel makes an angle \(\theta\) with the horizontal (see figure). At what time during a full revolution is \(\theta\) changing most rapidly?

Find the value of x that maximizes \(\theta\) in the figure.

Find the dimensions of the right circular cylinder of maximum volume that can be placed inside of a sphere of radius R.

Find the dimensions and area of the rectangle of maximum area that can be inscribed in the following figures. a. A right triangle with a given hypotenuse length L b. An equilateral triangle with a given side length L c. A right triangle with a given area A d. An arbitrary triangle with a given area A (The result applies to any triangle, but first consider triangles for which all the angles are less than or equal to \(90^{\circ}\).)

A right circular cylinder is placed inside a cone of radius R and height H so that the base of the cylinder lies on the base of the cone. a. Find the dimensions of the cylinder with maximum volume. Specifically, show that the volume of the maximum-volume cylinder is \(\frac{4}{9}\) the volume of the cone. b. Find the dimensions of the cylinder with maximum lateral surface area (area of the curved surface).

Suppose you own a tour bus and you book groups of 20 to 70 people for a day tour. The cost per person is $30 minus $0.25 for every ticket sold. If gas and other miscellaneous costs are $200, how many tickets should you sell to maximize your profit? Treat the number of tickets as a nonnegative real number.

A right circular cone is inscribed inside a larger right circular cone with a volume of \(150 \quad \mathrm{cm}^{3}\). The axes of the cones coincide and the vertex of the inner cone touches the center of the base of the outer cone. Find the ratio of the heights of the cones that maximizes the volume of the inner cone.

a. What point on the line y = 3x + 4 is closest to the origin? b. What point on the parabola \(y=1-x^{2}\) is closest to the point (1, 1)? c. Find the point on the graph of \(y=\sqrt{x}\) that is nearest the point (p, 0) if (i) \(p>\frac{1}{2}\); and (ii) \(0<p<\frac{1}{2}\).

a. What is the length of the longest pole that can be carried horizontally around a corner at which a 3-ft corridor and a 4-ft corridor meet at right angles? b. What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles? c. What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a = 5 ft wide and a corridor that is b = 5 ft wide meet at an angle of \(120^{\circ}\)? d. What is the length of the longest pole that can be carried around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles, assuming there is an 8-ft ceiling and that you may tilt the pole at any angle?

A load must be suspended 6 m below a high ceiling using cables attached to two supports that are 2 m apart (see figure). How far below the ceiling (x in the figure) should the cables be joined to minimize the total length of cable used?

The intensity of a light source at a distance is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. Two light sources, one twice as strong as the other, are 12 m apart. At what point on the line segment joining the sources is the intensity the weakest?

A rectangular sheet of paper of width a and length b, where 0 < a < b, is folded by taking one corner of the sheet and placing it at a point P on the opposite long side of the sheet (see figure). The fold is then flattened to form a crease across the sheet. Assuming that the fold is made so that there is no flap extending beyond the original sheet, find the point P that produces the crease of minimum length. What is the length of that crease?

An island is 3.5 mi from the nearest point on a straight shoreline; that point is 8 mi from a power station (see figure). A utility company plans to lay electrical cable underwater from the island to the shore and then underground along the shore to the power station. Assume that it costs $2400/mi to lay underwater cable and $1200/mi to lay underground cable. At what point should the underwater cable meet the shore in order to minimize the cost of the project?

Solve the problem in Exercise 31, but this time minimize the cost with respect to the smaller angle \(\theta\) between the underwater cable and the shore. (You should get the same answer.)

a. An isosceles triangle has a base of length 4 and two sides of length \(2 \sqrt{2}\). Let P be a point on the perpendicular bisector of the base. Find the location P that minimizes the sum of the distances between P and the three vertices. b. Assume in part (a) that the height of the isosceles triangle is h > 0 and its base has length 4. Show that the location of P that gives a minimum solution is independent of h for \(h \geq \frac{2}{\sqrt{3}}\).

What are the radius and area of the circle of maximum area that can be inscribed in an isosceles triangle whose two equal sides have length 1?

Among all right circular cones with a slant height of 3, what are the dimensions (radius and height) that maximize the volume of the cone? The slant height of a cone is the distance from the outer edge of the base to the vertex.

Suppose that a blood test for a disease must be given to a population of N people, where N is large. At most N individual blood tests must be done. The following strategy reduces the number of tests. Suppose 100 people are selected from the population and their blood samples are pooled. One test determines whether any of the 100 people test positive. If the test is positive, those 100 people are tested individually, making 101 tests necessary. However, if the pooled sample tests negative, then 100 people have been tested with one test. This procedure is then repeated. Probability theory shows that if the group size is x (for example, x = 100, as described here), then the average number of blood tests required to test N people is \(N\left(1-q^{x}+1 / x\right)\), where q is the probability that any one person tests negative. What group size x minimizes the average number of tests in the case that N = 10,000 and q = 0.95? Assume that x is a nonnegative real number.

A crank of radius r rotates with an angular frequency \(\omega\). It is connected to a piston by a connecting rod of length L (see figure). The acceleration of the piston varies with the position of the crank according to the function \(a(\theta)=\omega^{2} r\left(\cos \theta+\frac{r \cos 2 \theta}{L}\right)\). For fixed \(\omega\) and r, find the values of \(\theta\), with \(0 \leq \theta \leq 2 \pi\), for which the acceleration of the piston is a maximum and minimum.

A rain gutter is made from sheets of metal 9 in wide. The gutters have a 3-in base and two 3-in sides, folded up at an angle \(\theta\) (see figure). What angle \(\theta\) maximizes the cross-sectional area of the gutter?