Rectangles in triangles Find the dimensions and area of the rectangle of maximum area that can be inscribed in the following figures.
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° .)

Maximizing profit Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking people on a city tour is P(n) = n(50 - 0.5n) - 100. (Although P is defined only for positive integers, treat it as a continuous function.)

a. How many people should the guide take on a tour to maximize the profit?

{Use of Tech} Growth rate of spotted owlets The rate of growth (in g/week) of the body mass of Indian spotted owlets is modeled by the function r(t) = 10,147.9e⁻²·²ᵗ/(37.98e⁻²·² + 1), where t is the age (in weeks) of the owlets. What value of t > 0 maximizes r? What is the physical meaning of the maximum value?

{Use of Tech} Demand functions and elasticity Economists use demand functions to describe how much of a commodity can be sold at varying prices. For example, the demand function D(p) = 500 - 10p says that at a price of p = 10, a quantity of D(10) = 400 units of the commodity can be sold. The elasticity E = dD/dp p/D of the demand gives the approximate percent change in the demand for every 1% change in the price. (See Section 3.6 or the Guided Project Elasticity in Economics for more on demand functions and elasticity.)

b. If the price is $12 and increases by 4.5%, what is the approximate percent change in the demand? 

Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.

c. For arbitrary positive values of K and b, when does the maximum growth rate occur (in terms of K and b)?

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure).​​ <IMAGE>

a. Find the time at which the object first passes the rest position, y = 0. 

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure).​​ <IMAGE>

b. Find the time and the displacement when the object reaches its lowest point.

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure).​​ <IMAGE>

c. Find the time at which the object passes the rest position for the second time.

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure).​​ <IMAGE>

d. Find the time and the displacement when the object reaches its high point for the second time.

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.

Find the height h, radius r, and volume of a right circular cylinder with maximum volume that is inscribed in a sphere of radius R.

Rectangles beneath a semicircle 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?

Rectangles beneath a parabola A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 48 - x². What are the dimensions of the rectangle with the maximum area? What is the area?

Closest point on a curve What point on the parabola y = 1 - x² is closest to the point (1, 1)?

Minimum distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)?

Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.

a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.

Suppose S = x + 2y is an objective function subject to the constraint xy = 50, for x > 0 and y > 0.

b. Find the absolute minimum value of S subject to the given constraint.

Shortest ladder 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 to the house without touching the fence. Assume the vertical wall of the house and the horizontal ground have infinite extent.

Maximum-volume cone A cone is constructed by cutting a sector from a circular sheet of metal with radius 20. The cut sheet is then folded up and welded (see figure). Find the radius and height of the cone with maximum volume that can be formed in this way. <IMAGE>

Optimal soda can

a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm³ that minimize the surface area.

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

Rectangles beneath a line

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?

Folded boxes

a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 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.

Light transmission 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?

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

Maximum-area rectangles Of all rectangles with a fixed perimeter of P, which one has the maximum area? (Give the dimensions in terms of P.)

Minimum sum Find positive numbers x and y satisfying the equation xy = 12 such that the sum 2x + y is as small as possible.

Pen problems

b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 m² (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used? <IMAGE>

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs $p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns $w/hour.

e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.

Shipping crates A square-based, box-shaped shipping crate is designed to have a volume of 16 ft³. The material used to make the base costs twice as much (per square foot) as the material in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are the dimensions of the crate that minimize the cost of materials?

Snell’s Law Suppose a light source at A is in a medium in which light travels at a speed v₁ and that point B is in a medium in which light travels at a speed v₂ (see figure). Using Fermat’s Principle, which states that light travels along the path that requires the minimum travel time (Exercise 55), show that the path taken between points A and B satisfies (sinΘ₁/v₁ = (sin Θ₂) /v₂ . <IMAGE>

M​​inimizing related functions Complete each of the following parts.

b. What value of x minimizes ƒ(x) = (x- a₁)² + (x - a₂)² + (x - a₃)² , for constants a₁, a₂, and a₃?

Metal rain gutters 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 Θ (see figure). What angle Θ maximizes the cross-sectional area of the gutter? <IMAGE>

The arbelos 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). <IMAGE>

b. Show that the area of the arbelos is the area of a circle whose diameter is the distance BD in the figure.

Turning a corner with a pole

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?

Do dogs know calculus? 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 C is the point on the edge of the beach closest to the tennis ball (see figure). <IMAGE>

a. Assume the dog runs at speed r and swims at speed s, where r > s and both are measured in meters per second. 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. 

Viewi​​ng angles 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? <IMAGE>

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 Θ₁ = Θ₂ (see figure). <IMAGE>

Cylinder and cones (Putnam Exam 1938) 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?

Cylinder in a cone 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.

b. Find the dimensions of the cylinder with maximum lateral surface area (area of the curved surface).

Another pen problem 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). <IMAGE>

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.

Tree notch (Putnam Exam 1938, rephrased) A notch is cut in a cylindrical vertical tree trunk (see figure). 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 Θ. Prove that for a given tree and fixed angle Θ, the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.

Optimal popcorn box A small popcorn box is created from a 12" x 12" sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box? <IMAGE>

{Use of Tech } Min​​imizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs $p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns $w/hour.

b. At what speed does the gas mileage function have its maximum?

Optimization A right triangle has legs of length h and r and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = (1/3) πr²h.) <IMAGE>

Maxi​​mum printable area A rectangular page in a text (with width x and length y) has an area of 98 in² , top and bottom margins set at 1 in, and left and right margins set at 1/2 in. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?

Maximum area A line segment of length 10 joins the points (0, p) and (q, 0) to form a triangle in the first quadrant. Find the values of p and q that maximize the area of the triangle.

Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m³ needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.

A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of 32 ft³ , to be welded from quarter-inch plate, and to weigh no more than necessary. What dimensions do you recommend?

Your company can manufacture x hundred grade A tires and y hundred grade B tires a day, where 0 ≤ x ≤ 4 and y = (40 - 10x)/(5-x). Your profit on a grade A tire is twice your profit on a grade B tire. What is the most profitable number of each kind to make?

Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .

a. What is the farthest apart the particles ever get?

The ladder problem What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot.

A poster is set to have a total area of 1150 cm², with 2-cm margins on the sides and the top, and a 3-cm margin at the bottom. What dimensions will maximize the printed area?

Your café sells lattes for $4 each to 100 customers per day. For every $1 increase in price, you would lose 20 customers. Find the price that maximizes revenue. Hint: The # of items sold is based on the number of customers.