Finding the Box with the Largest Volume
Finding the dimensions of the box with the largest volume given the amount of material to make the box
tags:
Cost Function Critical Value Derivative Domain How-To Calculus Optimization Second Derivative Second Derivative TestMinimizing the Cost of a Box
Author: ShaNisaa RaSun
The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Answer" to reveal the answer to the problem.
Problem: A manufacturer is going to make rectangular storage containers with open tops. The volume of each container must be 10 cubic meters, and the length of each container will be twice its width. The cost of the material for the four sides of the container is \(\$4\) per square meter. The cost of the material for the bottom of the container is \(\$8\) per square meter. Find the manufacturer's minimum cost of making each container.
The manufacturer's minimum cost of making each container is \(\$115.86\).
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Using the Quotient and Chain Rules to find several derivatives
MLC WIR 20B M151 week6 #1gh
Using the Product and Chain Rules to differentiate functions with logarithms
MLC WIR 20B M151 week6 #1ij
Differentiating functions with logarithmic and inverse trigonometric functions
MLC WIR 20B M151 week6 #1kl
Differentiating functions with inverse trigonometric functions
MLC WIR 20B M151 week6 #2a
Finding the derivative of an implicit function
MLC WIR 20B M151 week6 #2b
Finding the derivative of an implicit function
MLC WIR 20B M151 week6 #3
Using the chain rule to find a derivative
MLC WIR 20B M151 week6 #4a
Using logarithmic differentiation to find the derivative of a function
MLC WIR 20B M151 week6 #4b
Using logarithmic differentiation to find the derivative of a function
MLC WIR 20B M151 week6 #5a
Using the Product Rule to find the derivative of a product of functions
MLC WIR 20B M151 week6 #5b
Using the Quotient Rule to find a derivative
MLC WIR 20B M151 week6 #5c
Using the Chain Rule to find the derivative of a composition of functions
MLC WIR 20B M151 week6 #17
Finding the linear approximation for a function
MLC WIR 20B M151 week6 #11
Using a linear approximation to approximate the value of a function
MLC WIR 20B M151 week7 #2a
Finding a number that satisfies the conclusion of the Mean Value Theorem on an interval
MLC WIR 20B M151 week7 #2b
Finding a number that satisfies the conclusion of the Mean Value Theorem on an interval
MLC WIR 20B M151 week7 #4a
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4b
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4c
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4d
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4e
Evaluating a limit with an indeterminate product using L'Hospital's Rule
MLC WIR 20B M151 week7 #4f
Evaluating a limit with an indeterminant difference using L'Hospital's Rule
MLC WIR 20B M151 week7 #4g
Evaluating a limit with an indeterminate power using L'Hospital's Rule
MLC WIR 20B M151 week7 #4h
Evaluating a limit with an indeterminant power using L'Hospital's Rule
MLC WIR 20B M151 week8 #5a
Finding a function from its second derivative using antidifferentiation
MLC WIR 20B M151 week8 #5b
Antidifferentiating twice to find a function from its second derivative
MLC WIR 20B M151 week8 #5c
Antidifferentiating twice to find a function from its second derivative
MLC WIR 20B M151 week8 #5e
Antidifferentiating twice to find a function from its second derivative
MLC WIR 20B M151 week8 #5f
Antidifferentiating twice to find a function from its second derivative
MLC WIR 20B M151 week8 #6
Using antidifferentiation to find the height of a cliff from the impact speed of a dropped stone
MLC WIR 20B M151 week9 #1
Determining the properties of a function from the graph of its derivative
MLC WIR 20B M151 week9 #8c
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week9 #8d
Evaluating a limit with an indeterminant power using L'Hospital's Rule
MLC WIR 20B M151 week9 #8e
Evaluating a limit with an indeterminant product using L'Hospital's Rule
MLC WIR 20B M151 week9 #9
Finding a number that satisfies the conclusion of the Mean Value Theorem for a function on an interval
MLC WIR 20B M151 week9 #8f
Evaluating a limit with an indeterminant product using L'Hospital's Rule
MLC WIR 20B M151 week9 #10a
Finding the antiderivative of a polynomial
MLC WIR 20B M151 week9 #10b
Finding the antiderivative of a rational function
MLC WIR 20B M151 week9 #10c
Finding the antiderivative of a function with secant and tangent
MLC WIR 20B M151 week9 #10d
Finding the antiderivative of a function and using a function value to find the constant
MLC WIR 20B M151 week10 #6c
Using L'Hospital's Rule to evaluate a limit representing a derivative at a point
MLC WIR 20B M151 week10 #9a
Finding a derivative using the Product Rule
MLC WIR 20B M151 week10 #9b
Finding a derivative using the Chain Rule multiple times
MLC WIR 20B M151 week10 #9c
Finding a derivative using the Chain Rule and Product Rule
MLC WIR 20B M151 week10 #9d
Finding a derivative using the Quotient Rule
MLC WIR 20B M151 week10 #10a
Finding the derivative of an implicit function
MLC WIR 20B M151 week10 #10b
Finding the derivative of an implicit function
MLC WIR 20B M151 week10 #11
Using logarithmic differentiation to find the derivative of a function
MLC WIR 20B M151 week10 #12
Finding the tangent line to a parametric curve
MLC WIR 20B M151 week10 #13
Finding when the tangent line to a parametric curve is vertical or horizontal
MLC WIR 20B M151 week10 #15
Determining the rate that the area of a triangle is increasing based on the rate an angle is increasing
MLC WIR 20B M151 week10 #16
Using a linear approximation to estimate the value of a function
MLC WIR 20B M151 week10 #17
Finding the value for a function that satisfies the Mean Value Theorem on an interval
MLC WIR 20B M151 week10 #18
Determining the properties of a function from the graph of its derivative
MLC WIR 20B M151 week10 #19a
Using L'Hospital's Rule to solve a limit
MLC WIR 20B M151 week10 #19b
Using L'Hospital's Rule to solve a limit with an indeterminant product
MLC WIR 20B M151 week10 #19c
Using L'Hospital's Rule to solve a limit with an indeterminant difference
MLC WIR 20B M151 week10 #19d
Using L'Hospital's Rule to solve a limit with an indeterminant power
WIR 20B M152 V88
Finding the length of a curve given by parametric equations
WIR 20B M152 V90
Finding the surface area of a rotated parametric curve
WIR 20B M152 V81
Finding the Taylor series of a function
WIR 20B M152 V82
Finding the Taylor series of a function
WIR 20B M152 V70
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V76
Finding a power series representation for a function
Parametric Curves: MATH 151 J.3 Problems 1-5
Cartesian equations and parametric equations of curves
Vectors and Derivatives: MATH 151 K.1 Problems 1-8
Derivatives and vectors with some physics applications
Vectors and Derivatives (Product Rule): MATH 171 Problems 1-3
Proving facts about the derivatives of vector functions including the product rule
Derivatives and Applications: MATH 171 Problems 7 & 8
Calculating slopes of tangent lines to parametric curves
Derivatives and Applications: MATH 151 K.2 Problems 9-15
Tangent lines to parametric equations and related rates examples
Review for the Common Exam: MATH 151 Exam 2 Review Problems 1-9
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Review for the Common Exam: MATH 151 Exam 2 Review Problems 21-29
Related rates problems, differentials, linear and quadratic approximations
Review for the Common Exam: MATH 151 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Review for the Common Exam: MATH 151 Exam 2 Review Problems 10-15
Implicit differentiation and physics applications of derivatives
Continuity: MATH 151 2.5 Problems 1-6
Continuity of Functions and the Intermediate Value Theorem
Derivatives and Rates of Change: MATH 171 Problems 1-3
Using the limit definition to find derivatives of functions and vector functions
Differentiation Formulas: MATH 171 Problems 4 & 5
Using the limit definition to find a derivative and proving the power rule
Differentiation Formulas: MATH 171 Problems 6 & 7
Proving the Product Rule and the rule for the derivative of a difference of functions
Review for the Common Exam: MATH 151 3.1 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Differentials and Approximations: MATH 151 3.10 Problems 1-6
Differentials, linear approximations and quadratic approximations
Differentials: MATH 171 Problems 1 & 2
Proving facts about differentials including a product rule
Review for the Common Exam: MATH 151 3.2 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Trigonometry Derivatives and the Chain Rule: MATH 151 3.3 Problems 1-9
Derivatives of trigonometric functions and using the Chain Rule
Trigonometry Derivatives and the Chain Rule: MATH 171 Problems 1-3
Proving the derivatives of trigonometric functions and that sine is continuous
Trigonometry Derivatives and the Chain Rule: MATH 171 Problems 4 & 5
Using the Chain Rule to prove facts about derivatives
Exponentials and Hyperbolic Trigonometric Functions: MATH 171 Problems 3 & 4
Examples involving the tangent line to an exponential function and finding the derivative of hyperbolic cosine
Trigonometry Derivatives and the Chain Rule: MATH 151 3.4 Problems 1-9
Derivatives of trigonometric functions and using the Chain Rule
Vectors and Derivatives: MATH 151 3.4 Problems 1-8
Derivatives and vectors with some physics applications
Implicit Differentiation: MATH 171 Problems 6 & 7
Using implicit differentiation to solve a problem and to prove the power rule for rational exponents
Implicit Differentiation: MATH 151 3.5 Problems 10-13
Implicit differentiation and finding tangent lines
Derivatives of Logarithmic Functions: MATH 171 Problems 1-3
Proving the derivative of ln(x) and the power and product rules using logarithmic differentiation
Differentiation and Applications: MATH 151 Problems 1-8
Derivatives of exponential and logarithmic functions and the exponential model
Derivatives and Applications: MATH 151 3.9 Problems 9-15
Tangent lines to parametric equations and related rates examples
Mean Value Theorem: MATH 171 Problems 6-8
Using the Mean Value Theorem to prove facts about functions
L'Hospital's Rule: MATH 171 Problems 7-9
Proving L'Hospital's Rule and using it to evaluate limits
Antiderivatives: MATH 171 Problems 4-6
Proving facts about antiderivatives and a physics application
Riemann Sums and Definite Integrals: MATH 151 5.2 Problems 6-12
Using Reimann sums and the Fundamental Theorem of Calculus
Riemann Sums and Definite Integrals: MATH 151 5.3 Problems 6-12
Using Riemann sums and the Fundamental Theorem of Calculus
Fundamental Theorem of Calculus: MATH 171 Problems 1-3
Proving and then applying the Fundamental Theorem of Calculus
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
Integration by Parts: MATH 172 Problems 1 & 2
Proving the formula for integration by parts and a reduction formula
WIR2 20B M251 V6
Determining the domain of a three-dimensional vector function
WIR2 20B M251 V8
Finding a vector equation for the tangent line to a three-dimensional vector function
WIR2 20B M251 V9
Finding the angle of intersection for two three-dimensional vector functions
WIR2 20B M251 V10
Finding the length of a three-dimensional curve
WIR3 20B M251 V1
Finding and sketching the domain of a function of two variables
WIR3 20B M251 V2
Finding and sketching the domain of a function of two variables
WIR4 20B M251 V9
Finding the absolute maximum and minimum values of a function of two variables on a rectangle
WIR4 20B M251 V10
Finding the absolute maximum and minimum values of a function of two variables on a triangular region
WIR8 20B M251 V1
Evaluating a line integral along half a circle
WIR8 20B M251 V2
Evaluating a line integral along a curve in space
WIR8 20B M251 V3
Evaluating a line integral along a curve
WIR8 20B M251 V4
Evaluating a line integral along a curve
WIR8 20B M251 V5
Evaluating a line integral along a curve in space
WIR8 20B M251 V6
Evaluating a line integral of a vector field along an curve
WIR8 20B M251 V7
Evaluating a line integral of a vector field along a curve in space
WIR8 20B M251 V8
Using a line integral to find the work done by a force field moving a particle along a curve
Matrix Derivative and Integral
Explaining how to find the derivative and integral of matrix and vector functions
MATH 152: Sequences Exercise 4
Finding the limit of a sequence
MATH 152: Sequences Exercise 6
Finding the limit of a sequence using logarithm properties
MATH 152: Sequences Exercise 7
Finding the limit of a sequence
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
MATH 152: Sequences Exercise 10
Determining if a sequence is increasing, decreasing, or non monotonic
WIR 20B M152 V45
Determining if a sequence converges or diverges
Functions of Several Variables Exercise V1
Evaluating and then finding and sketching the domain of a function of two variables
Functions of Several Variables Exercise V2
Finding and sketching the domain of a function of two variables
Functions of Several Variables Exercise V4
Sketching the level curves for a function of two variables
Functions of Several Variables Exercise V5
Sketching the level curves for a function of two variables
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
The Chain Rule (multivariable) Exercise V3
Using the Chain Rule for multivariable functions to find the rate a volume is changing
Maximum and Minimum Value (multivariable) Conceptual V2
Explaining the Second Derivative Test for Local Extrema for functions of two variables
Maximum and Minimum Value (multivariable) Exercise V1
Finding all local extrema and saddle points for a function of two variables
Maximum and Minimum Value (multivariable) Exercise V2
Finding all local extrema and saddle points for a function of two variables
Finding the nth Taylor Polynomial Exercise
Finding the nth Taylor polynomial of the natural log function centered at 1
Approximation Using Taylor Polynomials Exercise
Using a Taylor polynomial to approximate the value of a function and estimating the error
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
Maximum and Minimum Value (multivariable) Conceptual V2
Explaining the Second Derivative Test for Local Extrema for functions of two variables
Math 140 5.4 Exercise 4
Writing the domain of functions with radicals in interval notation
WIR2 20B M150 V7
Solving a rational equation and checking the solutions
WIR4 20B M150 V21
Solving an equation with logarithmic functions
WIR4 20B M150 V22
Solving an equation with logarithmic functions
WIR4 20B M150 V23
Solving an equation with logarithmic functions
WIR4 20B M150 V24
Solving an equation with logarithmic functions
WIR8 20B M150 V11
Solving an equation with logarithmic functions
Math 140 Exam 3 Review Exercise 1
Determining if a graph represents a function and finding the domain and range
Math 140 Exam 3 Review Exercise 4
Finding the properties of a quadratic function such as the vertex, axis of symmetry, intercepts, etc.
Math 140 Exam 3 Review Exercise 5
Finding the domain, intercepts, holes, and vertical asymptotes for a rational function
Math 140 Exam 3 Review Exercise 8
Finding the maximum profit for a word problem given the cost and price-demand equation
Math 140 Exam 3 Review Exercise 9
Finding the domain of a piecewise-defined function and evaluating it for given x-values
Math 140 Exam 3 Review Exercise 10
Finding properties of a polynomial function such as the degree, end behavior, y-intercept, etc.
Math 140 Exam 3 Review Exercise 11
Finding properties of a polynomial function such as the degree, end behavior, y-intercept, etc.
Math 140 Exam 3 Review Exercise 13
Stating the domain in interval notation of a function with radicals, fractions, and exponentials
Math 140 Exam 3 Review Exercise 24
Find the domain of a function in interval notation with a fraction and logarithm
Domain of a Rational Function
Finding the domain of a rational function
Domain of a Rational Function (2)
Finding the domain of a rational function
Domain of Function with Roots
Finding the domain of a function with a fraction and radicals
Domain of a Square Root Function
Finding the domain of a function with a quadratic inside a square root
Inverse of a Linear Function
Finding the inverse of a linear function and its domain
Inverse of a Square Root Function
Finding the inverse of a square root function and its domain
Inverse of a Rational Function
Finding the inverse of a rational function and its domain
Inverse of a Rational Function (2)
Finding the inverse of a rational function and its domain
Domain and x-intercepts of a Logarithmic Function
Finding the domain and x-intercepts of a logarithmic function
Domain and x-intercepts of a Logarithmic Function (2)
Finding the domain and x-intercepts of a logarithmic function
Solving an Equation with Logarithms
Solving a logarithmic equation using properties of logarithms
Solving an Equation with a Logarithmic Function
Solving a logarithmic equation using properties of logarithms
A Composition of Tangent and Arccosine
Evaluating a composition of tangent and arccosine
WIR 20B M152 V51
Determining if a series converges or diverges
WIR 20B M152 V63
Using the Alternating Series Test to determine if a series converges or diverges
Finding Absolute (Global) Extrema in Python
Using Python to find the absolute maximum and minimum of a function on a closed interval
Finding and Graphing Tangent Lines to a Curve in Python
Using Python to find the equation of the tangent line to a curve and graphing the result
Finding Derivatives in Python
Finding derivatives in Python and solving for the rate of change of a force
Finding Horizontal Tangent Lines in Python
Finding when the tangent line to a function is horizontal in Python
List Comprehension and Patterns in Higher Derivatives in Python
Using Python's list comprehension tool to find several higher order derivatives of a function at once
Parametric Plots and Tangents at a Given t-value in Python
Using Python to find the tangent line to a parametric equation and plot the two graphs
Tangents to Parametrized Curves at a Given Point in Python
Using Python to plot a parametrized curve and its tangent line at a point
Domain Restrictions
Explaining domain restrictions for denominators, even roots, and logarithms
Domain of Radical Functions Video 1
Finding the domain of even and odd roots
Domain of Radical Functions Video 2
Finding the domain of functions with even and odd roots
Domain of Rational Functions Video 1
Finding the domain of several rational functions
Domain of Rational Functions Video 2
Finding the domain of a rational function
Domain of Rational Functions Video 3
Explaining how to find any holes and vertical asymptotes of a rational function
Domain of Rational Functions Video 4
Finding the domain of a rational function using factoring
Domain of Logarithmic Functions Video 1
Finding the domain of a logarithmic function
Domain of Combinations of Functions Video 1
Finding the domain of a function with a denominator, square root, and logarithm
Domain of Combinations of Functions Video 2
Finding the domain of a function with a natural logarithm and denominator
Domain of Combinations of Functions Video 4
Finding the domain of a function with a root and a logarithm in the denominator
Linear Forms Video 2
Discussing properties of linear functions
Linear Graphs Video 1
Discussing the types of linear graphs including horizontal and vertical lines and their properties
Quadratic Form Video 1
Explaining the graph and properties for the parent function of quadratics
Quadratic Form Video 2
Discussing properties of quadratic functions including the vertex, domain, range, and end behavior
Properties of Rational Functions and Expressions Video 1
Discussing parent functions of rational functions and their properties
Properties of Rational Functions and Expressions Video 9
Explaining how to multiply and divide rational expressions
Properties of Rational Functions and Expressions Video 10
Performing algebraic operations with rational expressions and simplifying the answer
Solving Rational Equations Video 1
Solving equations that contain rational expressions
Solving Rational Equations Video 2
Solving an equation that contains rational expressions
Solving Rational Equations Video 3
Solving an equation that contains rational expressions
Properties of Radical Functions and Expressions Video 1
Explaining the general properties of even and odd root functions
Solving Radical Equations Video 1
Explaining how to solve radical equations and then solving example problems
Solving Radical Equations Video 2
Solving equations with even roots and checking the solutions
Solving Radical Equations Video 3
Solving equations with odd radicals
Solving Radical Equations Video 4
Solving equations with multiple radicals
Properties of Power and Exponential Expressions Video 4
Properties of exponential growth and decay functions
Properties of Power and Exponential Expressions Video 5
Properties of the exponential function with base e
Solving Power and Exponential Equations Video 1
Explaining how to solve equations with power and exponential terms
Properties of Logarithmic Functions and Expressions Video 2
Explaining the basic properties of logarithmic functions
Properties of Logarithmic Functions and Expressions Video 3
Expanding logarithmic expressions using logarithm rules
Properties of Logarithmic Functions and Expressions Video 4
Condensing a logarithmic expression into a single logarithm
Solving Logarithmic Equations Video 1
Explaining how to solve logarithmic equations
Solving Logarithmic Equations Video 2
Solving logarithmic equations with a single logarithm term
Solving Logarithmic Equations Video 3
Solving logarithmic equations with multiple logarithm terms
Solving Logarithmic Equations Video 4
Solving logarithmic equations with multiple logarithm terms on both sides
Splitting a Domain Into Two Pieces for a Double Integral
Evaluating a double integral over a given region by splitting it into two double integrals
Linear Forms Video 2
Discussing properties of linear functions
Linear Graphs Video 1
Discussing the types of linear graphs including horizontal and vertical lines and their properties
Properties of Power and Exponential Expressions Video 4
Properties of exponential growth and decay functions
Properties of Power and Exponential Expressions Video 5
Properties of the exponential function with base e
Properties of Logarithmic Functions and Expressions Video 2
Explaining the basic properties of logarithmic functions
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Maximum and Minimum Value (multivariable) Conceptual V2
Explaining the Second Derivative Test for Local Extrema for functions of two variables
Introduction to Riemann Sums
Estimating the area under a function using right-hand, left-hand, and midpoint Riemann sums
Review for the Common Exam: MATH 151 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative