Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.5.35.a
Capitolo 4, Problema 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.

Guida verificata passo dopo passo
1
Start by identifying the formulas involved: The volume of a cylinder is given by \( V = \pi r^2 h \) and the surface area is given by \( A = 2\pi r^2 + 2\pi rh \).
Given that the volume \( V = 354 \text{ cm}^3 \), express the height \( h \) in terms of the radius \( r \) using the volume formula: \( h = \frac{354}{\pi r^2} \).
Substitute \( h \) from the volume equation into the surface area formula to express the surface area \( A \) solely in terms of \( r \): \( A = 2\pi r^2 + \frac{708}{r} \).
To find the radius that minimizes the surface area, take the derivative of \( A \) with respect to \( r \), set it equal to zero, and solve for \( r \). This involves finding \( \frac{dA}{dr} = 4\pi r - \frac{708}{r^2} \) and setting \( \frac{dA}{dr} = 0 \).
Solve the equation \( 4\pi r - \frac{708}{r^2} = 0 \) to find the critical points. Then, use the second derivative test or analyze the behavior of \( \frac{dA}{dr} \) to confirm that the solution gives a minimum surface area. Finally, use the value of \( r \) to find \( h \) using the expression \( h = \frac{354}{\pi r^2} \).

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:
8m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Volume of a Cylinder

The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius and h is the height. In this problem, the volume is given as 354 cm³, which serves as a constraint for determining the optimal dimensions of the can.
Video consigliato:
08:29
Example 5: Packaging Design

Surface Area of a Cylinder

The surface area of a cylinder is given by the formula A = 2πr² + 2πrh, where the first term accounts for the areas of the two circular bases and the second term accounts for the lateral surface area. Minimizing this surface area while maintaining a fixed volume is the core objective of the problem.
Video consigliato:
09:07
Example 1: Minimizing Surface Area

Optimization Techniques

Optimization in calculus involves finding the maximum or minimum values of a function. In this case, techniques such as setting up a function for surface area in terms of one variable (using the volume constraint) and applying derivatives to find critical points are essential for solving the problem.
Video consigliato:
10:13
Intro to Applied Optimization: Maximizing Area
Pratica correlata
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
Domanda del libro di testo

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>

355
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

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

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.

268
views
Domanda del libro di testo

Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.


a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.

276
views