Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 70

Chapter 4, Problem 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.

Verified step by step guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

13m

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

Recommended video:

06:11

Fundamental Theorem of Calculus Part 1

Related Practice

Textbook Question

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.

Textbook Question

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.

Textbook Question

Theory and Examples

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

Textbook Question

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?

Textbook Question

Theory and Examples

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

Textbook Question

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.

views