Briggs - Calculus: Early Transcendentals 3rd Edition

Ch. 4 - Applications of the Derivative

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.5.35.a

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.

Problem 4.5.36

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?

Problem 4.5.38.a

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?

Problem 4.5.40.a

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.

Problem 4.5.42

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?

Problem 4.5.11

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

Problem 4.5.12

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.)

Problem 4.5.15

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

Problem 4.5.16.b

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>

Problem 4.5.21

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?

Problem 4.5.56

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>

Problem 4.5.61

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>

Problem 4.5.50a

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.

Problem 4.5.51

Viewing 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>

Problem 4.5.54b

{Use of Tech} Basketball shot A basketball is shot with an initial velocity of v ft/s at an angle of 45° to the floor. The center of the basketball is 8 ft above the floor at a horizontal distance of 18 feet from the center of the basketball hoop when it is released. The height h (in feet) of the center of the basketball after it has traveled a horizontal distance of x feet is modeled by the function h(x) = 32x² / v² + x + 8 (see figure). <IMAGE>

b. During the flight of the basketball, show that the distance s from the center of the basketball to the front of the hoop is s = √ (x - 17.25)² + ( -(4x² / 81) + x - 2)² (Hint: The diameter of the basketball hoop is 18 inches.)

Problem 4.5.55a

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>

Problem 4.5.68

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?

Problem 4.5.70d

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° .)

Problem 4.5.71b

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).

Problem 4.5.65

Crankshaft A crank of radius r rotates with an angular frequency w It is connected to a piston by a connecting rod of length L (see figure). The acceleration of the piston varies with the position of the crank according to the function <IMAGE>

a (Θ) = w²r (cos Θ + (r cos2Θ) / L) .

For fixed w , L, and r find the values of Θ, with 0 ≤ Θ ≤ 2π , for which the acceleration of the piston is a maximum and minimum.

Problem 4.5.77

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.

Problem 4.6.37

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

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Problem 4.6.53c

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. If f(x) = mx + b, then the linear approximation to f at any point is L(x) = f(x).

Problem 4.6.57

Approximating changes

Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).

Problem 4.6.59

Approximating changes

Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone of fixed height h = 6m when its radius decreases from r = 10 m to r = 9.9 m (S = πr√(r² + h²).