Solved: Sketch the graph of on the interval (a) Find the distance from the origin to the

Numerical, Graphical, and Analytic Analysis Find two positive numbers whose sum is 110 and whose product is a maximum. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the solution. (Hint: Use the table feature of the graphing utility.) (c) Write the product as a function of (d) Use a graphing utility to graph the function in part (c) and estimate the solution from the graph. (e) Use calculus to find the critical number of the function in part (c). Then find the two numbers

Numerical, Graphical, and Analytic Analysis An open box of maximum volume is to be made from a square piece of material, 24 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum volume. (b) Write the volume as a function of (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

In Exercises 3 8, find two positive numbers that satisfy the given requirements.The sum is and the product is a maximum.

In Exercises 3 8, find two positive numbers that satisfy the given requirements.The product is 192 and the sum is a minimum.

In Exercises 3 8, find two positive numbers that satisfy the given requirements.. The product is 192 and the sum of the first plus three times the second is a minimum.

In Exercises 3 8, find two positive numbers that satisfy the given requirements.The second number is the reciprocal of the first and the sum is a minimum.

In Exercises 3 8, find two positive numbers that satisfy the given requirements.The sum of the first and twice the second is 100 and the product is a maximum.

In Exercises 3 8, find two positive numbers that satisfy the given requirements.The sum of the first number squared and the second is 27 and the product is a maximum.

In Exercises 9 and 10, find the length and width of a rectangle that has the given perimeter and a maximum area.Perimeter: 100 meters

In Exercises 9 and 10, find the length and width of a rectangle that has the given perimeter and a maximum area.Perimeter: units

In Exercises 11 and 12, find the length and width of a rectangle that has the given area and a minimum perimeter. Area: 64 square feet

In Exercises 11 and 12, find the length and width of a rectangle that has the given area and a minimum perimeter. Area: square centimeters

In Exercises 13–16, find the point on the graph of the function that is closest to the given point. Function Point \(f(x)=\sqrt{x}\) (4, 0) Equation Transcription: Text Transcription: f(x) = sqrt x

In Exercises 13–16, find the point on the graph of the function that is closest to the given point. Function Point \(f(x)=\sqrt{x-8}\) (2, 0) Equation Transcription: Text Transcription: f(x) = sqrt x - 8

In Exercises 13–16, find the point on the graph of the function that is closest to the given point. Function Point \(f(x)=x^{2}\) \(\left(2, \frac{1}{2}\right)\) Equation Transcription: Text Transcription: f(x) = x^2 (2, 1/2)

In Exercises 13–16, find the point on the graph of the function that is closest to the given point. Function Point \(f(x)=(x+1)^{2}\) (5, 3) Equation Transcription: Text Transcription: f(x) = (x+1)^2

Chemical Reaction In an autocatalytic chemical reaction, the product formed is a catalyst for the reaction. If is the amount of the original substance and is the amount of catalyst formed, the rate of chemical reaction is For what value of will the rate of chemical reaction be greatest?

Traffic Control On a given day, the flow rate (in cars per hour) on a congested roadway is where is the speed of the traffic in miles per hour. What speed will maximize the flow rate on the road?

Area A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

Maximum Area A rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals (see figure). What dimensions should be used so that the enclosed area will be a maximum?

Maximum Volume (a) Verify that each of the rectangular solids shown in the figure has a surface area of 150 square inches. (b) Find the volume of each solid. (c) Determine the dimensions of a rectangular solid (with a square base) of maximum volume if its surface area is 150 square inches.

Maximum Volume Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 337.5 square centimeters.

Maximum Area A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.

Maximum Area A rectangle is bounded by the and axes and the graph of (see figure). What length and width should the rectangle have so that its area is a maximum?

Minimum Length A right triangle is formed in the first quadrant by the and axes and a line through the point (see figure). (a) Write the length of the hypotenuse as a function of (b) Use a graphing utility to approximate graphically such that the length of the hypotenuse is a minimum. (c) Find the vertices of the triangle such that its area is a minimum.

Maximum Area Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 4 (see figure). (a) Solve by writing the area as a function of (b) Solve by writing the area as a function of (c) Identify the type of triangle of maximum area.

Maximum Area A rectangle is bounded by the axis and the semicircle (see figure). What length and width should the rectangle have so that its area is a maximum?

Area Find the dimensions of the largest rectangle that can be inscribed in a semicircle of radius (see Exercise 27).

Area A rectangular page is to contain 30 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Area A rectangular page is to contain 36 square inches of print. The margins on each side are to be inches. Find the dimensions of the page such that the least amount of paper is used

Numerical, Graphical, and Analytic Analysis An exercise room consists of a rectangle with a semicircle on each end. A 200-meter running track runs around the outside of the room. (a) Draw a figure to represent the problem. Let and represent the length and width of the rectangle. (b) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum area of the rectangular region. (c) Write the area as a function of (d) Use calculus to find the critical number of the function in part (c) and find the maximum value. (e) Use a graphing utility to graph the function in part (c) and verify the maximum area from the graph.

Numerical, Graphical, and Analytic Analysis A right circular cylinder is to be designed to hold 22 cubic inches of a soft drink (approximately 12 fluid ounces). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the minimum surface area. (Hint: Use the table feature of the graphing utility.) (c) Write the surface area as a function of (d) Use a graphing utility to graph the function in part (c) and estimate the minimum surface area from the graph. (e) Use calculus to find the critical number of the function in part (c) and find dimensions that will yield the minimum surface area

Maximum Volume A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)

Maximum Volume Rework Exercise 33 for a cylindrical package. (The cross section is circular.)

Maximum Volume Find the volume of the largest right circular cone that can be inscribed in a sphere of radius r

Maximum Volume Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius r

The perimeter of a rectangle is 20 feet. Of all possible dimensions, the maximum area is 25 square feet when its length and width are both 5 feet. Are there dimensions that yield a minimum area? Explain.

A shampoo bottle is a right circular cylinder. Because the surface area of the bottle does not change when it is squeezed, is it true that the volume remains the same? Explain.

Minimum Surface Area A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

Minimum Cost An industrial tank of the shape described in Exercise 39 must have a volume of 3000 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize cost

Minimum Area The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.

Maximum Area Twenty feet of wire is to be used to form two figures. In each of the following cases, how much wire should be used for each figure so that the total enclosed area is a maximum? (a) Equilateral triangle and square (b) Square and regular pentagon (c) Regular pentagon and regular hexagon (d) Regular hexagon and circle What can you conclude from this pattern? {Hint: The area of a regular polygon with sides of length is A
n4cot
n x2.

Beam Strength A wooden beam has a rectangular cross section of height and width (see figure on the next page). The strength of the beam is directly proportional to the width and the square of the height. What are the dimensions of the strongest beam that can be cut from a round log of diameter 24 inches? (Hint: where is the proportionality constant.)

Minimum Length Two factories are located at the coordinates and with their power supply located at (see figure). Find such that the total length of power line from the power supply to the factories is a minimum.

Projectile Range The range of a projectile fired with an initial velocity at an angle with the horizontal is where is the acceleration due to gravity. Find the angle such that the range is a maximum.

Conjecture Consider the functions and on the domain (a) Use a graphing utility to graph the functions on the specified domain. (b) Write the vertical distance between the functions as a function of and use calculus to find the value of for which is maximum. (c) Find the equations of the tangent lines to the graphs of and at the critical number found in part (b). Graph the tangent lines. What is the relationship between the lines? (d) Make a conjecture about the relationship between tangent lines to the graphs of two functions at the value of at which the vertical distance between the functions is greatest, and prove your conjecture.

Illumination A light source is located over the center of a circular table of diameter 4 feet (see figure). Find the height of the light source such that the illumination at the perimeter of the table is maximum if where is the slant height, is the angle at which the light strikes the table, and is a constant.

Illumination The illumination from a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. Two light sources of intensities and are units apart. What point on the line segment joining the two sources has the least illumination?

Minimum Time A man is in a boat 2 miles from the nearest point on the coast. He is to go to a point located 3 miles down the coast and 1 mile inland (see figure). He can row at 2 miles per hour and walk at 4 miles per hour. Toward what point on the coast should he row in order to reach point in the least time?

Minimum Time Consider Exercise 49 if the point is on the shoreline rather than 1 mile inland. (a) Write the travel time as a function of (b) Use the result of part (a) to find the minimum time to reach (c) The man can row at miles per hour and walk at miles per hour. Write the time as a function of Show that the critical number of depends only on and and not on the distances. Explain how this result would be more beneficial to the man than the result of Exercise 49. (d) Describe how to apply the result of part (c) to minimizing the cost of constructing a power transmission cable that costs dollars per mile under water and dollars per mile over land.

Minimum Time The conditions are the same as in Exercise 49 except that the man can row at miles per hour and walk at miles per hour. If and are the magnitudes of the angles, show that the man will reach point in the least time when sin 1 v1 sin 2 v2 .

Minimum Time When light waves, traveling in a transparent medium, strike the surface of a second transparent medium, they change direction. This change of direction is called refraction and is defined by Snells Law of Refraction, where and are the magnitudes of the angles shown in the figure and and are the velocities of light in the two media. Show that this problem is equivalent to Exercise 51, and that light waves traveling from to follow the path of minimum time

Sketch the graph of on the interval (a) Find the distance from the origin to the intercept and the distance from the origin to the intercept. (b) Write the distance from the origin to a point on the graph of as a function of Use a graphing utility to graph and find the minimum distance. (c) Use calculus and the zero or root feature of a graphing utility to find the value of that minimizes the function on the interval What is the minimum distance? (Submitted by Tim Chapell, Penn Valley Community College, Kansas City, MO.)

Minimum Cost An offshore oil well is 2 kilometers off the coast. The refinery is 4 kilometers down the coast. Laying pipe in the ocean is twice as expensive as on land. What path should the pipe follow in order to minimize the cost?

Maximum Volume A sector with central angle is cut from a circle of radius 12 inches (see figure), and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of the cone is a maximum.

Numerical, Graphical, and Analytic Analysis The cross sections of an irrigation canal are isosceles trapezoids of which three sides are 8 feet long (see figure). Determine the angle of elevation of the sides such that the area of the cross section is a maximum by completing the following. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) (b) Use a graphing utility to generate additional rows of the table and estimate the maximum cross-sectional area. (Hint: Use the table feature of the graphing utility.) (c) Write the cross-sectional area as a function of (d) Use calculus to find the critical number of the function in part (c) and find the angle that will yield the maximum cross-sectional area. (e) Use a graphing utility to graph the function in part (c) and verify the maximum cross-sectional area.

Minimum Cost The ordering and transportation cost of the components used in manufacturing a product is where is measured in thousands of dollars and is the order size in hundreds. Find the order size that minimizes the cost. (Hint: Use the root feature of a graphing utility.)

Diminishing Returns The profit (in thousands of dollars) for a company spending an amount (in thousands of dollars) on advertising is (a) Find the amount of money the company should spend on advertising in order to obtain a maximum profit. (b) The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline. Find the point of diminishing returns.

Area Find the area of the largest rectangle that can be inscribed under the curve in the first and second quadrants.

Population Growth Fifty elk are introduced into a game preserve. It is estimated that their population will increase according to the model where is measured in years. At what rate is the population increasing when After how many years is the population increasing most rapidly?

Verify that the function increases at the maximum rate when y L2.

Area Perform the following steps to find the maximum area of the rectangle shown in the figure. (a) Solve for in the equation (b) Use the result in part (a) to write the area as a function of (c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions of the rectangle of maximum area. Determine the required area. (d) Use a graphing utility to graph the expression for found in part (a). Use the graph to approximate and Use this result to describe the changes in the dimensions and position of the rectangle for x 1 1 2 3 4 4 5 6 c c + x f(x) = 10xex y 0 < x < .

Minimum Distance In Exercises 63–65, consider a fuel distribution center located at the origin of the rectangular coordinate system (units in miles; see figures). The center supplies three factories with coordinates (4, 1), (5, 6), and (10, 3). A trunk line will run from the distribution center along the line y = mx, and feeder lines will run to the three factories. The objective is to find m such that the lengths of the feeder lines are minimized. Minimize the sum of the squares of the lengths of vertical feeder lines given by \(S_{1}=(4 m-1)^{2}+(5 m-6)^{2}+(10 m-3)^{2}\). Find the equation for the trunk line by this method and then determine the sum of the lengths of the feeder lines. Equation Transcription: Text Transcription: S_1 = (4m - 1)^2 + (5m - 6)^2 + (10m - 3)^2

Minimum Distance In Exercises 63–65, consider a fuel distribution center located at the origin of the rectangular coordinate system (units in miles; see figures). The center supplies three factories with coordinates (4, 1), (5, 6), and (10, 3). A trunk line will run from the distribution center along the line y = mx, and feeder lines will run to the three factories. The objective is to find m such that the lengths of the feeder lines are minimized. Minimize the sum of the absolute values of the lengths of vertical feeder lines given by \(S_{2}=|4 m-1|+|5 m-6|+|10 m-3|\). Find the equation for the trunk line by this method and then determine the sum of the lengths of the feeder lines. (Hint: Use a graphing utility to graph the function and approximate the required critical number.) Equation Transcription: Text Transcription: S_2 = |4m - 1| + |5m - 6| + |10m - 3|

Minimum Distance In Exercises 63–65, consider a fuel distribution center located at the origin of the rectangular coordinate system (units in miles; see figures). The center supplies three factories with coordinates (4, 1), (5, 6), and (10, 3). A trunk line will run from the distribution center along the line y = mx, and feeder lines will run to the three factories. The objective is to find m such that the lengths of the feeder lines are minimized. Minimize the sum of the perpendicular distances (see Exercises 85–90 in Section 1.2) from the trunk line to the factories given by \(S_{3}=\frac{|4 m-1|}{\sqrt{m^{2}+1}}+\frac{|5 m-6|}{\sqrt{m^{2}+1}}+\frac{|10 m-3|}{\sqrt{m^{2}+1}}\). Find the equation for the trunk line by this method and then determine the sum of the lengths of the feeder lines. (Hint: Use a graphing utility to graph the function \(S_{3}\) and approximate the required critical number.) Equation Transcription: Text Transcription: S_3 = |4m - 1| / sqrt m^2 + 1 + |5m - 6| / sqrt m^2 + 1 + |10m - 3| / sqrt m^2 + 1

Maximum Area Consider a symmetric cross inscribed in a circle of radius (see figure). (a) Write the area of the cross as a function of and find the value of that maximizes the area. (b) Write the area of the cross as a function of and find the value of that maximizes the area. (c) Show that the critical numbers of parts (a) and (b) yield the same maximum area. What is that area?

Find the maximum value of on the set of all real numbers satisfying Explain your reasoning.

Find the minimum value of for These problems were composed by the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved. x > 0.
x  1x6
x6  1x6 2
x  1x3 
x