A rectangular beam is cut from a cylindrical log of radius 30 cm. The strength of a beam

If a differentiable function f(x) has a global maximum on the interval 0 x 10 at x = 0, then f (0) 0.

The sum of two nonnegative numbers is 100. What is the maximum value of the product of these two numbers?

The product of two positive numbers is 784. What is the minimum value of their sum?

The sum of three nonnegative numbers is 36, and one of the numbers is twice one of the other numbers. What is the maximum value of the product of these three numbers?

The perimeter of a rectangle is 64 cm. Find the lengths of the sides of the rectangle giving the maximum area.

If you have 100 feet of fencing and want to enclose a rectangular area up against a long,

A closed rectangular box, with a square base x by x cm and height h cm.

An open-topped rectangular box, with a square base x by x cm and height h cm.

A closed cylinder with radius r cm and height h cm.

A closed cylinder with radius r cm and height h cm.

In Exercises 1011, find the x-value maximizing the shaded area. One vertex is on the graph of f(x) = x2/350x+1000, where 0 x 20.

In Exercises 1011, find the x-value maximizing the shaded area. One vertex is on the graph of f(x) = x2/350x+1000, where 0 x 20.

A rectangle has one side on the x-axis and two vertices on the curve y = 1 1 + x2 .

A right triangle has one vertex at the origin and one vertex on the curve y = ex/3 for 1 x 5. One of the two perpendicular sides is along the x-axis; the other is parallel to the y-axis. Find

A rectangle has one side on the x-axis, one side on the y-axis, one vertex at the origin and one on the curve y = e2x for x 0. Find the (a) Maximum area (b) Minimum perimeter

Find analytically the exact critical point of the function which represents the time, t, to walk to the bus stop in Example 2. Recall that t is given by t = x 6 + (2000 x)2 + 6002 4 .

Of all rectangles with given area, A, which has the shortest diagonals

A rectangular beam is cut from a cylindrical log of radius 30 cm. The strength of a beam of width w and height h is proportional to wh2. (See Figure 4.40.) Find the width and height of the beam of maximum strength.

In Problems 1819 a vertical line divides a region into two pieces. Find the value of the coordinate x that maximizes the product of the two areas.

In Problems 1819 a vertical line divides a region into two pieces. Find the value of the coordinate x that maximizes the product of the two areas.

In Problems 2022 the figures are made of rectangles and semicircles. (a) Find a formula for the area. (b) Find a formula for the perimeter. (c) Find the dimensions x and y that maximize the area given that the perimeter is 100.

In Problems 2022 the figures are made of rectangles and semicircles. (a) Find a formula for the area. (b) Find a formula for the perimeter. (c) Find the dimensions x and y that maximize the area given that the perimeter is 100.

In Problems 2022 the figures are made of rectangles and semicircles. (a) Find a formula for the area. (b) Find a formula for the perimeter. (c) Find the dimensions x and y that maximize the area given that the perimeter is 100.

A piece of wire of length L cm is cut into two pieces. One piece, of length x cm, is made into a circle; the rest is made into a square. (a) Find the value of x that makes the sum of the areas of the circle and square a minimum. Find the value of x giving a maximum. (b) For the values of x found in part (a), show that the ratio of the length of wire in the square to the length of wire in the circle is equal to the ratio of the area of the square to the area of the circle.3 (c) Are the values of x found in part (a) the only values of x for which the ratios in part (b) are equal?

In Problems 2427, find the minimum and maximum values of the expression where x and y are lengths in Figure 4.41 and 0 x 10.

In Problems 2427, find the minimum and maximum values of the expression where x and y are lengths in Figure 4.41 and 0 x 10.

In Problems 2427, find the minimum and maximum values of the expression where x and y are lengths in Figure 4.41 and 0 x 10.

In Problems 2427, find the minimum and maximum values of the expression where x and y are lengths in Figure 4.41 and 0 x 10.

Which point on the curve y = 1 x is closest to the origin?

Find the point(s) on the ellipse x2 9 + y2 = 1 (a) Closest to the point (2, 0). (b) Closest to the focus ( 8, 0). [Hint: Minimize the square of the distancethis avoids

What are the dimensions of the closed cylindrical can that has surface area 280 square centimeters and contains the maximum volume?

A hemisphere of radius 1 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the cylinder of maximum volume.

In a chemical reaction, substance A combines with substance B to form substance Y . At the start of the reaction, the quantity of A present is a grams, and the quantity of B present is b grams. At time t seconds after the start of the reaction, the quantity of Y present is y grams. Assume a

A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks 20 miles apart, the concentration of the combined deposits on the line joining them, at a distance x from one stack, is given by where and are positive constants which depend on the quantity of smoke each stack is emitting. If , find the point on the line joining the stacks where the concentration of the deposit is a minimum.

A wave of wavelength traveling in deep water has speed, v, given for positive constants c and k, by v = k c + c As varies, does such a wave have a maximum or minimum velocity? If so, what is it? Explain.

A circular ring of wire of radius r0 lies in a plane perpendicular to the x-axis and is centered at the origin. The ring has a positive electric charge spread uniformly over it. The electric field in the x-direction, E, at the point x on the axis is given by E = kx (x2 + r2 0) 3/2 for k > 0. At what point on the x-axis is the field greatest? Least?

a mass of m kg. If her arm makes an angle of with her body (assumed vertical) and the coefficient of friction (a positive constant) is , the least force, F, she must exert to move the sled is given by F = mg sin + cos . If = 0.15, find the maximum and minimum values of F for 0 /2. Give answers as multiples of mg.

Four equally massive particles can be made to rotate, equally spaced, around a circle of radius r. This is physically possible provided the radius and period T of the rotation are chosen so that the following action function is at its global minimum: A(r) = r2 T + T r , r> 0. (a) Find the radius r at which A(r) has a global minimum. (b) If the period of the rotation is doubled, determine whether the radius of the rotation increases or decreases, and by approximately what percentage

Four equally massive particles can be made to rotate, equally spaced, around a circle of radius r. This is physically possible provided the radius and period T of the rotation are chosen so that the following action function is at its global minimum: A(r) = r2 T + T r , r> 0. (a) Find the radius r at which A(r) has a global minimum. (b) If the period of the rotation is doubled, determine whether the radius of the rotation increases or decreases, and by approximately what percentage

Four equally massive particles can be made to rotate, equally spaced, around a circle of radius r. This is physically possible provided the radius and period T of the rotation are chosen so that the following action function is at its global minimum: A(r) = r2 T + T r , r> 0. (a) Find the radius r at which A(r) has a global minimum. (b) If the period of the rotation is doubled, determine whether the radius of the rotation increases or decreases, and by approximately what percentage

A business sells an item at a constant rate of r units per month. It reorders in batches of q units, at a cost of a+bq dollars per order. Storage costs are k dollars per item per month, and, on average, q/2 items are in storage, waiting to be sold. [Assume r, a, b, k are positive constants.] (a) How often does the business reorder? (b) What is the average monthly cost of reordering? (c) What is the total monthly cost, C of ordering and storage? (d) Obtain Wilsons lot size formula, the optimal batch size which minimizes cost.

A bird such as a starling feeds worms to its young. To collect worms, the bird flies to a site where worms are to be found, picks up several in its beak, and flies back to its nest. The loading curve in Figure 4.42 shows how the number of worms (the load) a starling collects depends on the time it has been searching for them.4 The curve is concave down because the bird can pick up worms more efficiently when its beak is empty; when its beak is partly full, the bird becomes much less efficient. The traveling time (from nest to site and back) is represented by the distance P O in Figure 4.42. The bird wants to maximize the rate at which it brings worms to the nest, where Rate worms arrive = Load Traveling time + Searching time (a) Draw a line in Figure 4.42 whose slope is this rate. (b) Using the graph, estimate the load which maximizes this rate. (c) If the traveling time is increased, does the optimal load increase or decrease? Why? O P 4 8 time load (number of worms) Number of worms Traveling time Searching ti

On the same side of a straight river are two towns, and the townspeople want to build a pumping station, S. See Figure 4.43. The pumping station is to be at the rivers edge with pipes extending straight to the two towns. Where should the pumping station be located to minimize the total length of pipe?

A pigeon is released from a boat (point B in Figure 4.44) floating on a lake. Because of falling air over the cool water, the energy required to fly one meter over the lake is twice the corresponding energy e required for flying over the bank (e = 3 joule/meter). To minimize the energy required to fly from B to the loft, L, the pigeon heads to a point P on the bank and then flies along the bank to L. The distance AL is 2000 m, and AB is 500 m. (a) Express the energy required to fly from B to L via P as a function of the angle (the angle BPA). (b) What is the optimal angle ? (c) Does your answer change if AL, AB, and e have different numerical values?

To get the best view of the Statue of Liberty in Figure 4.45, you should be at the position where is a maximum. If the statue stands 92 meters high, including the pedestal, which is 46 meters high, how far from the base should you be? [Hint: Find a formula for in terms of your distance from the base. Use this function to maximize , noting that 0 /2.]

A light ray starts at the origin and is reflected off a mirror along the line y = 1 to the point (2, 0). See Figure 4.46. Fermats Principle says that lights path minimizes the time of travel.5 The speed of light is a constant. (a) Using Fermats Principle, find the optimal position of P. (b) Using your answer to part (a), derive the Law of Reflection, that 1 = 2.

(a) For which positive number x is x1/x largest? Justify your answer. [Hint: You may want to write x1/x = eln(x1/x) .] (b) For which positive integer n is n1/n largest? Justify your answer. (c) Use your answer to parts (a) and (b) to decide which is larger: 31/3 or 1/

The arithmetic mean of two numbers a and b is defined as (a+b)/2; the geometric mean of two positive numbers a and b is defined as ab. (a) For two positive numbers, which of the two means is larger? Justify your answer. [Hint: Define f(x)=(a+x)/2ax for fixed a.] (b) For three positive numbers a, b, c, the arithmetic and geometric mean are (a+b+c)/3 and 3 abc, respectively. Which of the two means of three numbers is larger? [Hint: Redefine f(x) for fixed a and b.]

A line goes through the origin and a point on the curve y = x2e3x, for x 0. Find the maximum slope of such a line. At what x-value does it occur?

The distance, s, traveled by a cyclist, who starts at 1 pm, is given in Figure 4.47. Time, t, is in hours since noon. (a) Explain why the quantity s/t is represented by the slope of a line from the origin to the point (t, s) on the graph. (b) Estimate the time at which the quantity s/t is a maximum. (c) What is the relationship between the quantity s/t and the instantaneous speed of the cyclist at the time you found in part (b)

When birds lay eggs, they do so in clutches of several at a time. When the eggs hatch, each clutch gives rise to a brood of baby birds. We want to determine the clutch size which maximizes the number of birds surviving to adulthood per brood. If the clutch is small, there are few baby birds in the brood; if the clutch is large, there are so many baby birds to feed that most die of starvation. The number of surviving birds per brood as a function of clutch size is shown by the benefit curve in Figure 4.48.6 (a) Estimate the clutch size which maximizes the number of survivors per brood. (b) Suppose also that there is a biological cost to having a larger clutch: the female survival rate is reduced by large clutches. This cost is represented by the dotted line in Figure 4.48. If we take cost into account by assuming that the optimal clutch size in fact maximizes the vertical distance between the curves, what is the new optimal clutch size?

Let f(v) be the amount of energy consumed by a flying bird, measured in joules per second (a joule is a unit of energy), as a function of its speed v (in meters/sec). Let a(v) be the amount of energy consumed by the same bird, measured in joules per meter. (a) Suggest a reason in terms of the way birds fly for the shape of the graph of f(v) in Figure 4.49. (b) What is the relationship between f(v) and a(v)? (c) Where on the graph is a(v) a minimum?

The forward motion of an aircraft in level flight is reduced by two kinds of forces, known as induced drag and parasite drag. Induced drag is a consequence of the downward deflection of air as the wings produce lift. Parasite drag results from friction between the air and the entire surface of the aircraft. Induced drag is inversely proportional to the square of speed and parasite drag is directly proportional to the square of speed. The sum of induced drag and parasite drag is called total drag. The graph in Figure 4.50 shows a certain aircrafts induced drag and parasite drag functions. (a) Sketch the total drag as a function of air speed. (b) Estimate two different air speeds which each result in a total drag of 1000 pounds. Does the total drag function have an inverse? What about the induced and parasite drag functions? (c) Fuel consumption (in gallons per hour) is roughly proportional to total drag. Suppose you are low on fuel and the control tower has instructed you to enter a circular holding pattern of indefinite duration to await the passage of a storm at your landing field. At what air speed should you fly the holding pattern? Why?

Let f(v) be the fuel consumption, in gallons per hour, of a certain aircraft as a function of its airspeed, v, in miles per hour. A graph of f(v) is given in Figure 4.51. (a) Let g(v) be the fuel consumption of the same aircraft, but measured in gallons per mile instead of gallons per hour. What is the relationship between f(v) and g(v)? (b) For what value of v is f(v) minimized? (c) For what value of v is g(v) minimized? (d) Should a pilot try to minimize f(v) or g(v)?

If A is the area of a rectangle of sides x and 2x, for 0 x 10, the maximum value of A occurs where dA/dx = 0.

An open box is made from a 20-inch square piece of cardboard by cutting squares of side h from the corners and folding up the edges, giving the box in Figure 4.52. To find the maximum volume of such a box, we work on the domain h 0

The solution of an optimization problem modeled by a quadratic function occurs at the vertex of the quadratic.

The solution of an optimization problem modeled by a quadratic function occurs at the vertex of the quadratic.

A context for a modeling problem where you are given that xy = 120 and you are minimizing the quantity 2x + 6y

A modeling problem where you are minimizing the cost of the material in a cylindrical can of volume 250 cubic centimeters.