Answer: Consider the following problem: A box with an open top is to be constructed from

Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in your table, estimate the answer to the problem. (b) Use calculus to solve the problem and compare with your answer to part (a).

Find two numbers whose difference is 100 and whose product is a minimum.

Find two positive numbers whose product is 100 and whose sum is a minimum.

The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?

What is the maximum vertical distance between the line and the parabola for ?

What is the minimum vertical distance between the parabolas and ?

Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible

Find the dimensions of a rectangle with area whose perimeter is as small as possible.

. A model used for the yield of an agricultural crop as a function of the nitrogen level in the soil (measured in appropriate units) is where is a positive constant. What nitrogen level gives the best yield?

The rate at which photosynthesis takes place for a species of phytoplankton is modeled by the function where is the light intensity (measured in thousands of footcandles). For what light intensity is a maximum?

Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then 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? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. Does it appear that there is a maximum volume? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the volume. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the volume as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).

A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?

A box with a square base and open top must have a volume of 32,000 cm . Find the dimensions of the box that mini mize the amount of material used.

If 1200 cm of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

A rectangular storage container with an open top is to have a volume of 10 m . 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.

Do Exercise 16 assuming the container has a lid that is made from the same material as the sides.

(a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.

Find the point on the line that is closest to the origin.

Find the point on the curve that is closest to the point .

Find the points on the ellipse that are farthest away from the point .

Find, correct to two decimal places, the coordinates of the point on the curve that is closest to the point .

Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius .

Find the area of the largest rectangle that can be inscribed in the ellipse .

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side if one side of the rectangle lies on the base of the triangle.

Find the area of the largest trapezoid that can be inscribed in a circle of radius 1 and whose base is a diameter of the circle.

Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius .

Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs.

A right circular cylinder is inscribed in a sphere of radius . Find the largest possible volume of such a cylinder.

A right circular cylinder is inscribed in a cone with height and base radius . Find the largest possible volume of such a cylinder.

A right circular cylinder is inscribed in a sphere of radius . Find the largest possible surface area of such a cylinder.

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 62 on page 22.) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.

The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm , find the dimensions of the poster with the smallest area.

A poster is to have an area of 180 in 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?

A piece of wire 10 m 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) a maximum? (b) A minimum?

Answer Exercise 35 if one piece is bent into a square and the other into a circle.

A cylindrical can without a top is made to contain of liquid. Find the dimensions that will minimize the cost of the metal to make the can

A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

A cone-shaped drinking cup is made from a circular piece of paper of radius by cutting out a sector and joining the edges and . Find the maximum capacity of such a cup

A cone-shaped paper drinking cup is to be made to hold of water. Find the height and radius of the cup that will use the smallest amount of paper.

A cone with height is inscribed in a larger cone with height so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when .

An object with weight is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with a plane, then the magnitude of the force is where is a constant called the coefficient of friction. For what value of is smallest?

If a resistor of ohms is connected across a battery of volts with internal resistance ohms, then the power ( in watts) in the external resistor is If and are fixed but varies, what is the maximum value of the power?

For a fish swimming at a speed relative to the water, the energy expenditure per unit time is proportional to . It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current , then the time required to swim a distance is and the total energy required to swim the distance is given by where is the proportionality constant. (a) Determine the value of that minimizes . (b) Sketch the graph of . Note: This result has been verified experimentally; migrating fish swim against a current at a speed greater than the current speed.

In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to minimize the surface area for a given volume, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area is given by where , the length of the sides of the hexagon, and , the height, are constants. (a) Calculate . (b) What angle should the bees prefer? (c) Determine the minimum surface area of the cell ( in terms of and ). Note: Actual measurements of the angle in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than .

A boat leaves a dock at 2:00 PM and travels due south at a speed of 20 kmh. Another boat has been heading due east at 15 kmh and reaches the same dock at 3:00 PM. At what time were the two boats closest together?

. Solve the problem in Example 4 if the river is 5 km wide and point is only 5 km downstream from .

A woman at a point on the shore of a circular lake with radius 2 mi wants to arrive at the point diametrically opposite on the other side of the lake in the shortest possible time (see the figure). She can walk at the rate of 4 mih and row a boat at 2 mih. How should she proceed? 49

An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is over land to a point on the north bank and under the river to the tanks. To minimize the cost of the pipeline, where should be located?

Suppose the refinery in Exercise 49 is located 1 km north of the river. Where should be located?

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 ft apart, where should an object be placed on the line between the sources so as to receive the least illumination?

Find an equation of the line through the point that cuts off the least area from the first quadrant.

Let and be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point .

At which points on the curve does the tangent line have the largest slope?

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve at some point?

What is the smallest possible area of the triangle that is cut off by the first quadrant and whose hypotenuse is tangent to the parabola at some point?

(a) If is the cost of producing units of a commodity, then the average cost per unit is . Show that if the average cost is a minimum, then the marginal cost equals the average cost.(b) If , in dollars, find (i) the cost, average cost, and marginal cost at a production level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the minimum average cost.

(a) Show that if the profit is a maximum, then the marginal revenue equals the marginal cost. (b) If is the cost function and is the demand function, find the production level that will maximize profit.

A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at , the average attendance had been 27,000. When ticket prices were lowered to , the average attendance rose to 33,000. (a) Find the demand function, assuming that it is linear. (b) How should ticket prices be set to maximize revenue?

During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for each and his sales averaged 20 per day. When he increased the price by , he found that the average decreased by two sales per day. (a) Find the demand function, assuming that it is linear. (b) If the material for each necklace costs Terry , what should the selling price be to maximize his profit?

A manufacturer has been selling 1000 flat-screen TVs a week at each. A market survey indicates that for each rebate offered to the buyer, the number of TVs sold will increase by 100 per week. (a) Find the demand function. (b) How large a rebate should the company offer the buyer in order to maximize its revenue? (c) If its weekly cost function is , how should the manufacturer set the size of the rebate in order to maximize its profit?

The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is per month. A market survey suggests that, on average, one additional unit will remain vacant for each increase in rent. What rent should the manager charge to maximize revenue?

Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.

The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be?

A point needs to be located somewhere on the line so that the total length of cables linking to the points , , and is minimized (see the figure). Express as a function of and use the graphs of and to estimate the minimum value of .

The graph shows the fuel consumption of a car (measured in gallons per hour) as a function of the speed of the car. At very low speeds the engine runs inefficiently, so initially decreases as the speed increases. But at high speeds the fuel consumption increases. You can see that is minimized for this car when mih. However, for fuel efficiency, what must be minimized is not the consumption in gallons per hour but rather the fuel consumption in gallons per mile. Lets call this consumption . Using the graph, estimate the speed at which has its minimum value.

Let be the velocity of light in air and the velocity of light in water. According to Fermats Principle, a ray of light will travel from a point in the air to a point in the water by a path that minimizes the time taken. Show that where (the angle of incidence) and (the angle of refrac tion) are as shown. This equation is known as Snells Law.

Two vertical poles and are secured by a rope going from the top of the first pole to a point on the ground R between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when .

The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose to minimize ?

A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?

An observer stands at a point , one unit away from a track. Two runners start at the point in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observers angle of sight between the runners. [Hint: Maximize .]

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 . How should be chosen so that the gutter will carry the maximum amount of water?

Where should the point be chosen on the line segment so as to maximize the angle ?

A painting in an art gallery has height and is hung so that its lower edge is a distance above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle subtended at his eye by the painting?)

Find the maximum area of a rectangle that can be circum - scribed about a given rectangle with length and width . [Hint: Express the area as a function of an angle .]

The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuilles Laws gives the resistance of the blood as where is the length of the blood vessel, is the radius, and is a positive constant determined by the viscosity of the blood. (Poiseuille established this law experimentally, but it also follows from Equation 8.4.2.) The figure shows a main blood ves sel with radius branching at an angle into a smaller vessel with radius . (a) Use Poiseuilles Law to show that the total resistance of the blood along the path is where and are the distances shown in the figure. (b) Prove that this resistance is minimized when (c) Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is twothirds the radius of the larger vessel.

Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5 km from the nearest point on a straight shoreline, flies to a point on the shoreline, and then flies along the shoreline to its nesting area . Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points and are 13 km apart. (a) In general, if it takes 1.4 times as much energy to fly over water as it does over land, to what point should the bird fly in order to minimize the total energy expended in returning to its nesting area? (b) Let and L denote the energy (in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio WL mean in terms of the birds flight? What would a small value mean? Determine the ratio corresponding to the minimum expenditure of energy. (c) What should the value of be in order for the bird to fly directly to its nesting area ? What should the value of be for the bird to fly to and then along the shore to ? (d) If the ornithologists observe that birds of a certain species reach the shore at a point 4 km from B, how many times more energy does it take a bird to fly over water than over land?

Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point on a line parallel to the line joining the light sources and at a distance meters from it (see the figure). We want to locate on so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. (a) Find an expression for the intensity at the point . (b) If m, use graphs of and to show that the intensity is minimized when m, that is, when is at the midpoint of . (c) If m, show that the intensity (perhaps surpris - ingly) is not minimized at the midpoint. (d) Somewhere between m and m there is a transitional value of at which the point of minimal illumination abruptly changes. Estimate this value of by graphical methods. Then find the exact value of . P