Solved: Let v1 be the velocity of light in air and v2 the velocity of light in water

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 one at the right, 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). First number Second number Product 1 22 22 2 21 42 3 20 60 . . . . . .

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 y x 1 2 and the parabola y x 2 for 21 < x < 2?

What is the minimum vertical distance between the parabolas y x 2 1 1 and y x 2 x 2 ?

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 1000 m2 whose perimeter is as small as possible.

A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is Y kN 1 1 N2 where k is a positive constant. What nitrogen level gives the best yield?

The rate sin mg carbonym3 yhd at which photosynthesis takes place for a species of phytoplankton is modeled by the function P 100I I 2 1 I 1 4 where I is the light intensity (measured in thousands of footcandles). For what light intensity is P 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 in 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 cm3 . Find the dimensions of the box that minimize the amount of material used.

If 1200 cm2 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 m3 . 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 farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $20 per linear foot to install and the farmer is not willing to spend more than $5000, find the dimensions for the plot that would enclose the most area.

If the farmer in Exercise 18 wants to enclose 8000 square feet of land, what dimensions will minimize the cost of the fence?

(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 y 2x 1 3 that is closest to the origin.

Find the point on the curve y sx that is closest to the point s3, 0d.

Find the points on the ellipse 4x 2 1 y 2 4 that are farthest away from the point s1, 0d.

Find, correct to two decimal places, the coordinates of the point on the curve y sin x that is closest to the point s4, 2d.

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

Find the area of the largest rectangle that can be inscribed in the ellipse x 2 ya2 1 y 2 yb2 1.

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L 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 r.

If the two equal sides of an isosceles triangle have length a, find the length of the third side that maximizes the area of the triangle.

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

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

A right circular cylinder is inscribed in a sphere of radius r. 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 1.1.62.) 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 cm2 , find the dimensions of the poster with the smallest area.

A poster is to have an area of 180 in2 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 37 if one piece is bent into a square and the other into a circle.

If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 32 inches, what diameter pizza will reward you with the largest slice?

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 R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup. A B R C

A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.

A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h 1 3 H.

An object with weight W 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 F W  sin  1 cos  where  is a constant called the coefficient of friction. For what value of  is F smallest?

If a resistor of R ohms is connected across a battery of E volts with internal resistance r ohms, then the power (in watts) in the external resistor is P E2 R sR 1 rd 2 If E and r are fixed but R varies, what is the maximum value of the power?

For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v 3 . 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 u su , vd, then the time required to swim a distance L is Lysv 2 ud and the total energy E required to swim the distance is given by Esvd av 3 ? L v 2 u where a is the proportionality constant. (a) Determine the value of v that minimizes E. (b) Sketch the graph of E. Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50% 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 side length and height, 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 S is given by S 6sh 2 3 2 s2 cot  1 (3s 2 s3y2) csc  where s, the length of the sides of the hexagon, and h, the height, are constants. (a) Calculate dSyd. (b) What angle should the bees prefer? (c) Determine the minimum surface area of the cell (in terms of s and h). 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 28. s trihedral angle rear of cell front of cell h

A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 kmyh. Another boat has been heading due east at 15 kmyh 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 B is only 5 km downstream from A.

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

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 $400,000ykm over land to a point P on the north bank and $800,000ykm under the river to the tanks. To minimize the cost of the pipeline, where should P be located?

Suppose the refinery in Exercise 51 is located 1 km north of the river. Where should P 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 s3, 5d that cuts off the least area from the first quadrant.

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

At which points on the curve y 1 1 40x 3 2 3x 5 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 y 3yx 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 y 4 2 x 2 at some point?

(a) If Csxd is the cost of producing x units of a commodity, then the average cost per unit is csxd Csxdyx. Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If Csxd 16,000 1 200x 1 4x 3y2 , 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 Psxd is a maximum, then the marginal revenue equals the marginal cost. (b) If Csxd 16,000 1 500x 2 1.6x 2 1 0.004x 3 is the cost function and psxd 1700 2 7x 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 $10, the average attendance had been 27,000. When ticket prices were lowered to $8, 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 $10 each and his sales averaged 20 per day. When he increased the price by $1, 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 $6, what should the selling price be to maximize his profit?

A retailer has been selling 1200 tablet computers a week at $350 each. The marketing department estimates that an additional 80 tablets will sell each week for every $10 that the price is lowered. (a) Find the demand function. (b) What should the price be set at in order to maximize revenue? (c) If the retailers weekly cost function is Csxd 35,000 1 120x what price should it choose in order to maximize its profit?

A company operates 16 oil wells in a designated area. Each pump, on average, extracts 240 barrels of oil daily. The company can add more wells but every added well reduces the average daily ouput of each of the wells by 8 barrels. How many wells should the company add in order to maximize daily production?

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

Consider the situation in Exercise 51 if the cost of laying pipe under the river is considerably higher than the cost of laying pipe over land ($400,000ykm). You may suspect that in some instances, the minimum distance possible under the river should be used, and P should be located 6 km from the refinery, directly across from the storage tanks. Show that this is never the case, no matter what the under river cost is.

Consider the tangent line to the ellipse x 2 a2 1 y2 b2 1 at a point s p, qd in the first quadrant. (a) Show that the tangent line has x-intercept a2 yp and y-intercept b2 yq. (b) Show that the portion of the tangent line cut off by the coordinate axes has minimum length a 1 b. (c) Show that the triangle formed by the tangent line and the coordinate axes has minimum area ab.

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 a b b

A point P needs to be located somewhere on the line AD so that the total length L of cables linking P to the points A, B. and C is minimized (see the figure). Express L as a function of x | AP | and use the graphs of L and dLydx to estimate the minimum value of L. B C P A 2 m 3 m D 5 m

The graph shows the fuel consumption c of a car (measured in gallons per hour) as a function of the speed v of the car. At very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. But at high speeds the fuel consumption increases. You can see that csvd is minimized for this car when v < 30 miyh. 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 G. Using the graph, estimate the speed at which G has its minimum value. c 0 20 40 60

Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermats Principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken. Show that sin 1 sin 2 v1 v2 where 1 (the angle of incidence) and 2 (the angle of refraction) are as shown. This equation is known as Snells Law. C A B

Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground 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 1 2. Q R T P S

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 x to minimize y? x y 8 12

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? 6 9

An observer stands at a point P, one unit away from a track. Two runners start at the point S 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. S 1 P

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? 10 cm 10 cm 10 cm

Where should the point P be chosen on the line segment AB so as to maximize the angle ? 5 2 A B P

A painting in an art gallery has height h and is hung so that its lower edge is a distance d 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?) h d

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W. [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 R of the blood as R C L r 4 where L is the length of the blood vessel, r is the radius, and C 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 vessel with radius r1 branching at an angle  into a smaller vessel with radius r2. b A B r r C a vascular branching (a) Use Poiseuilles Law to show that the total resistance of the blood along the path ABC is R CS a 2 b cot  r1 4 1 b csc  r2 4 D where a and b are the distances shown in the figure. (b) Prove that this resistance is minimized when cos  r 4 2 r 4 1 (c) Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is two-thirds 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 B on a straight shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D 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 C should the bird fly in order to minimize the total energy expended in returning to its nesting area? (b) Let W and L denote the energy (in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio WyL mean in terms of the birds flight? What would a small value mean? Determine the ratio WyL corresponding to the minimum expenditure of energy. (c) What should the value of WyL be in order for the bird to fly directly to its nesting area D? What should the value of WyL be for the bird to fly to B and then along the shore to D? (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? 13 km B C D island 5 km nest

Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line ,, parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P 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 Isxd at the point P. (b) If d 5 m, use graphs of Isxd and I9sxd to show that the intensity is minimized when x 5 m, that is, when P is at the midpoint of ,. (c) If d 10 m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint. (d) Somewhere between d 5 m and d 10 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Estimate this value of d by graphical methods. Then find the exact value of d. P d 10 m