Domanda del libro di testo
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.
52
views
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.R.35
Optimal popcorn box A small popcorn box is created from a 12" x 12" sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box? <IMAGE>
Guida verificata passo dopo passo1
First, understand the problem: We need to maximize the volume of a box formed by cutting and folding a 12" x 12" sheet of paperboard. The cuts are four rectangles, each with dimensions x by x/2.
Calculate the dimensions of the base of the box after the cuts. The original sheet is 12" x 12". After cutting out the rectangles, the length and width of the base will be reduced by 2x each, resulting in dimensions (12 - 2x) by (12 - x).
The height of the box will be x/2, as this is the width of the rectangles being cut out and folded up to form the sides of the box.
Write the volume function V(x) for the box. The volume V is given by the product of the base area and the height: V(x) = (12 - 2x)(12 - x)(x/2).
To find the value of x that maximizes the volume, take the derivative of V(x) with respect to x, set it equal to zero, and solve for x. This will give the critical points. Use the second derivative test or analyze the sign changes to determine which critical point gives the maximum volume.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
7mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Volume of a Box
The volume of a rectangular box is calculated using the formula V = length × width × height. In this problem, the dimensions of the box are influenced by the cuts made from the paperboard, which will affect the final dimensions after folding. Understanding how to express the volume in terms of a single variable, based on the cuts made, is crucial for maximizing the volume.
Video consigliato:
08:29
Example 5: Packaging Design
Optimization
Optimization in calculus involves finding the maximum or minimum values of a function. In this context, we need to determine the dimensions that maximize the volume of the box. This typically involves taking the derivative of the volume function, setting it to zero to find critical points, and using the second derivative test to confirm whether these points yield a maximum volume.
Video consigliato:
10:13Intro to Applied Optimization: Maximizing Area
Derivatives
Derivatives represent the rate of change of a function and are fundamental in finding local maxima and minima. In this problem, we will differentiate the volume function with respect to the variable x, which represents the dimensions of the cuts. Analyzing the derivative helps identify critical points where the volume could be maximized, guiding us to the optimal dimensions for the box.
Video consigliato:
Percorso guidato
05:44Derivatives
Pratica correlata
Domanda del libro di testo
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
a. Find the critical points of f and determine where f is increasing and where it is decreasing.
235
views
Domanda del libro di testo
{Use of Tech } Minimizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)
281
views
Domanda del libro di testo
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?
212
views
Domanda del libro di testo
Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.
a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?
333
views
Domanda del libro di testo
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 4cos (π (x-1)) on [0, 2]
253
views