Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 76b
Chapter 4, Problem 76b

Turning a corner with a pole
What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles?

Verified step by step guidance
1
First, visualize the problem: You have two corridors meeting at a right angle, one is 'a' feet wide and the other is 'b' feet wide. You need to find the longest pole that can be carried horizontally around this corner.
Consider the pole as a line segment that needs to navigate the corner. The pole will touch the inner corner and the outer walls of the corridors. The key is to find the maximum length of this line segment.
Use the concept of similar triangles and geometry. The pole will form a right triangle with the walls of the corridors. The hypotenuse of this triangle is the pole, and the legs are the distances from the corner to the point where the pole touches the walls.
Set up the equation using the Pythagorean theorem. If the pole touches the walls at points (x, 0) and (0, y), then the equation is: x2+y2. You need to express x and y in terms of 'a' and 'b'.
To maximize the length of the pole, use calculus to find the critical points of the function derived from the geometry. This involves taking the derivative of the function with respect to x or y, setting it to zero, and solving for the maximum length.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
11m
Was this helpful?

Key Concepts

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

Geometric Optimization

Geometric optimization involves finding the maximum or minimum values of a function subject to certain constraints. In this problem, we need to determine the longest pole that can fit around a corner, which requires analyzing the geometric relationship between the pole and the corridors' widths.
Recommended video:
10:13
Intro to Applied Optimization: Maximizing Area

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is essential for calculating the effective length of the pole as it turns the corner, allowing us to relate the widths of the corridors to the pole's length.
Recommended video:
06:11
Fundamental Theorem of Calculus Part 1

Calculus of Variations

Calculus of variations is a field of mathematical analysis that deals with optimizing functionals, which are mappings from a set of functions to real numbers. In this context, it helps in determining the optimal shape or path that the pole must take to maximize its length while navigating the corner formed by the two corridors.
Recommended video:
06:11
Fundamental Theorem of Calculus Part 1
Related Practice
Textbook Question

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.


g(t) = 3t⁵ - 30t⁴ + 80t³ + 100

1
views
Textbook Question

Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.

lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)

Textbook Question

More limits Evaluate the following limits.

lim_x→1 (x ln x - x + 1) / (xln²x)

Textbook Question

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.


f(x) = 2x⁴ + 8x³ + 12x² - x - 2

Textbook Question

The arbelos An arbelos is the region enclosed by three mutually tangent semicircles; it is the region inside the larger semicircle and outside the two smaller semicircles (see figure). <IMAGE>

b. Show that the area of the arbelos is the area of a circle whose diameter is the distance BD in the figure.

Textbook Question

{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.


b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.