 4.7.1: Consider the following problem: Find two numbers whose sum is 23 an...
 4.7.2: Find two numbers whose difference is 100 and whose product is a min...
 4.7.3: Find two positive numbers whose product is 100 and whose sum is a m...
 4.7.4: The sum of two positive numbers is 16. What is the smallest possibl...
 4.7.5: What is the maximum vertical distance between the line y x 1 2 and ...
 4.7.6: What is the minimum vertical distance between the parabolas y x 2 1...
 4.7.7: Find the dimensions of a rectangle with perimeter 100 m whose area ...
 4.7.8: Find the dimensions of a rectangle with area 1000 m2 whose perimete...
 4.7.9: A model used for the yield Y of an agricultural crop as a function ...
 4.7.10: The rate sin mg carbonym3 yhd at which photosynthesis takes place f...
 4.7.11: Consider the following problem: A farmer with 750 ft of fencing wan...
 4.7.12: Consider the following problem: A box with an open top is to be con...
 4.7.13: A farmer wants to fence in an area of 1.5 million square feet in a ...
 4.7.14: A box with a square base and open top must have a volume of 32,000 ...
 4.7.15: If 1200 cm2 of material is available to make a box with a square ba...
 4.7.16: A rectangular storage container with an open top is to have a volum...
 4.7.17: Do Exercise 16 assuming the container has a lid that is made from t...
 4.7.18: A farmer wants to fence in a rectangular plot of land adjacent to t...
 4.7.19: If the farmer in Exercise 18 wants to enclose 8000 square feet of l...
 4.7.20: (a) Show that of all the rectangles with a given area, the one with...
 4.7.21: Find the point on the line y 2x 1 3 that is closest to the origin.
 4.7.22: Find the point on the curve y sx that is closest to the point s3, 0d.
 4.7.23: Find the points on the ellipse 4x 2 1 y 2 4 that are farthest away ...
 4.7.24: Find, correct to two decimal places, the coordinates of the point o...
 4.7.25: Find the dimensions of the rectangle of largest area that can be in...
 4.7.26: Find the area of the largest rectangle that can be inscribed in the...
 4.7.27: Find the dimensions of the rectangle of largest area that can be in...
 4.7.28: Find the area of the largest trapezoid that can be inscribed in a c...
 4.7.29: Find the dimensions of the isosceles triangle of largest area that ...
 4.7.30: If the two equal sides of an isosceles triangle have length a, find...
 4.7.31: A right circular cylinder is inscribed in a sphere of radius r. Fin...
 4.7.32: A right circular cylinder is inscribed in a cone with height h and ...
 4.7.33: A right circular cylinder is inscribed in a sphere of radius r. Fin...
 4.7.34: A Norman window has the shape of a rectangle surmounted by a semici...
 4.7.35: The top and bottom margins of a poster are each 6 cm and the side m...
 4.7.36: A poster is to have an area of 180 in2 with 1inch margins at the b...
 4.7.37: A piece of wire 10 m long is cut into two pieces. One piece is bent...
 4.7.38: Answer Exercise 37 if one piece is bent into a square and the other...
 4.7.39: If you are offered one slice from a round pizza (in other words, a ...
 4.7.40: A fence 8 ft tall runs parallel to a tall building at a distance of...
 4.7.41: A coneshaped drinking cup is made from a circular piece of paper o...
 4.7.42: A coneshaped paper drinking cup is to be made to hold 27 cm3 of wa...
 4.7.43: A cone with height h is inscribed in a larger cone with height H so...
 4.7.44: An object with weight W is dragged along a horizontal plane by a fo...
 4.7.45: If a resistor of R ohms is connected across a battery of E volts wi...
 4.7.46: For a fish swimming at a speed v relative to the water, the energy ...
 4.7.47: In a beehive, each cell is a regular hexagonal prism, open at one e...
 4.7.48: A boat leaves a dock at 2:00 pm and travels due south at a speed of...
 4.7.49: Solve the problem in Example 4 if the river is 5 km wide and point ...
 4.7.50: A woman at a point A on the shore of a circular lake with radius 2 ...
 4.7.51: An oil refinery is located on the north bank of a straight river th...
 4.7.52: Suppose the refinery in Exercise 51 is located 1 km north of the ri...
 4.7.53: The illumination of an object by a light source is directly proport...
 4.7.54: Find an equation of the line through the point s3, 5d that cuts off...
 4.7.55: Let a and b be positive numbers. Find the length of the shortest li...
 4.7.56: At which points on the curve y 1 1 40x 3 2 3x 5 does the tangent li...
 4.7.57: What is the shortest possible length of the line segment that is cu...
 4.7.58: What is the smallest possible area of the triangle that is cut off ...
 4.7.59: (a) If Csxd is the cost of producing x units of a commodity, then t...
 4.7.60: (a) Show that if the profit Psxd is a maximum, then the marginal re...
 4.7.61: A baseball team plays in a stadium that holds 55,000 spectators. Wi...
 4.7.62: During the summer months Terry makes and sells necklaces on the bea...
 4.7.63: A retailer has been selling 1200 tablet computers a week at $350 ea...
 4.7.64: A company operates 16 oil wells in a designated area. Each pump, on...
 4.7.65: Show that of all the isosceles triangles with a given perimeter, th...
 4.7.66: Consider the situation in Exercise 51 if the cost of laying pipe un...
 4.7.67: Consider the tangent line to the ellipse x 2 a2 1 y2 b2 1 at a poin...
 4.7.68: The frame for a kite is to be made from six pieces of wood. The fou...
 4.7.69: A point P needs to be located somewhere on the line AD so that the ...
 4.7.70: The graph shows the fuel consumption c of a car (measured in gallon...
 4.7.71: Let v1 be the velocity of light in air and v2 the velocity of light...
 4.7.72: Two vertical poles PQ and ST are secured by a rope PRS going from t...
 4.7.73: The upper righthand corner of a piece of paper, 12 in. by 8 in., a...
 4.7.74: A steel pipe is being carried down a hallway 9 ft wide. At the end ...
 4.7.75: An observer stands at a point P, one unit away from a track. Two ru...
 4.7.76: A rain gutter is to be constructed from a metal sheet of width 30 c...
 4.7.77: Where should the point P be chosen on the line segment AB so as to ...
 4.7.78: A painting in an art gallery has height h and is hung so that its l...
 4.7.79: Find the maximum area of a rectangle that can be circumscribed abou...
 4.7.80: The blood vascular system consists of blood vessels (arteries, arte...
 4.7.81: Ornithologists have determined that some species of birds tend to a...
 4.7.82: Two light sources of identical strength are placed 10 m apart. An o...
Solutions for Chapter 4.7: Optimization Problems
Full solutions for Single Variable Calculus: Early Transcendentals  8th Edition
ISBN: 9781305270336
Solutions for Chapter 4.7: Optimization Problems
Get Full SolutionsSingle Variable Calculus: Early Transcendentals was written by and is associated to the ISBN: 9781305270336. This textbook survival guide was created for the textbook: Single Variable Calculus: Early Transcendentals, edition: 8. Since 82 problems in chapter 4.7: Optimization Problems have been answered, more than 107079 students have viewed full stepbystep solutions from this chapter. Chapter 4.7: Optimization Problems includes 82 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Arccosecant function
See Inverse cosecant function.

Cofunction identity
An identity that relates the sine, secant, or tangent to the cosine, cosecant, or cotangent, respectively

Cone
See Right circular cone.

De Moivre’s theorem
(r(cos ? + i sin ?))n = r n (cos n? + i sin n?)

Domain of a function
The set of all input values for a function

Heron’s formula
The area of ¢ABC with semiperimeter s is given by 2s1s  a21s  b21s  c2.

Histogram
A graph that visually represents the information in a frequency table using rectangular areas proportional to the frequencies.

Inverse variation
See Power function.

Mathematical induction
A process for proving that a statement is true for all natural numbers n by showing that it is true for n = 1 (the anchor) and that, if it is true for n = k, then it must be true for n = k + 1 (the inductive step)

Onetoone function
A function in which each element of the range corresponds to exactly one element in the domain

Orthogonal vectors
Two vectors u and v with u x v = 0.

Pseudorandom numbers
Computergenerated numbers that can be used to approximate true randomness in scientific studies. Since they depend on iterative computer algorithms, they are not truly random

Right triangle
A triangle with a 90° angle.

Row operations
See Elementary row operations.

Scatter plot
A plot of all the ordered pairs of a twovariable data set on a coordinate plane.

Standard representation of a vector
A representative arrow with its initial point at the origin

Symmetric matrix
A matrix A = [aij] with the property aij = aji for all i and j

Triangular form
A special form for a system of linear equations that facilitates finding the solution.

Vertical line test
A test for determining whether a graph is a function.

Whole numbers
The numbers 0, 1, 2, 3, ... .