Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 70

Capitolo 4, Problema 70

The ladder problem What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot.

Guida verificata passo dopo passo

Visualize the problem as a right-angled triangle where the ladder forms the hypotenuse, and the two legs are the widths of the corridors. Let the widths of the corridors be 'a' and 'b'.

Use the Pythagorean theorem to express the length of the ladder (hypotenuse) in terms of 'a' and 'b'. The equation is:

c = \sqrt{a^{2} + b^{2}}

To find the maximum length of the ladder that can be carried horizontally, consider the geometry of the corner. The ladder must touch both walls at the corner, forming a right triangle with the corner.

Set up an equation for the ladder's length in terms of the angle it makes with one of the walls. Use trigonometric identities to express the length in terms of sine and cosine:

L = \frac{a}{\sin θ} + \frac{b}{\cos θ}

Differentiate the expression for 'L' with respect to the angle

θ

and find the critical points to determine the maximum length of the ladder. Use the first or second derivative test to confirm the maximum.

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:

13m

Concetti chiave

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

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In the context of the ladder problem, we need to determine the longest ladder that can fit around a corner, which requires setting up a function that describes the relationship between the ladder's length and the dimensions of the corridor.

Video consigliato:

10:13

Intro to Applied Optimization: Maximizing Area

Geometric Relationships

Understanding geometric relationships is crucial for solving the ladder problem. This involves visualizing the ladder as a hypotenuse of a right triangle formed by the corridor's walls and applying the Pythagorean theorem to relate the lengths of the sides to the ladder's length.

Video consigliato:

Percorso guidato

06:29

Derivatives Applied To Velocity

Calculus of Functions

The calculus of functions, particularly derivatives, is essential for finding critical points where the maximum length of the ladder occurs. By differentiating the function that models the ladder's length with respect to its position, we can identify the optimal length that satisfies the constraints of the corridor.

Video consigliato:

Percorso guidato

06:11

Fundamental Theorem of Calculus Part 1

Pratica correlata

Domanda del libro di testo

Let ƒ(x) = 3x - x³ . Show that the equation ƒ(𝓍) = -4 has a solution in the interval [2,3] and use Newton’s method to find it.

187

views

Domanda del libro di testo

Sketch the graph of a twice-differentiable function y=f(x) that passes through the points (-2,2), (-1,1), (0,0),(1,1), and (2,2) and whose first two derivatives have the following sign patterns.

224

views

Domanda del libro di testo

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local minima at (1, 1) and (3, 3).

212

views

Domanda del libro di testo

Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .

b. When do the particles collide?

914

views

Domanda del libro di testo

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local maxima at (1, 1) and (3, 3)

204

views

Domanda del libro di testo

Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.

225

views