Ch. 4 - Applications of Derivatives

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

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.5.45

Chapter 4, Problem 4.5.45

The 8-ft wall shown here stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

Verified step by step guidance

Visualize the problem as a right triangle where the beam is the hypotenuse, the wall is one leg, and the distance from the wall to the building is the other leg.

Let the point where the beam touches the wall be (x, 8) and the point where it touches the building be (27, y). The beam forms a right triangle with the ground and the wall.

The length of the beam can be expressed using the Pythagorean theorem: \( L = \sqrt{x^2 + (y - 8)^2} \).

To minimize the length of the beam, we need to express y in terms of x using the similar triangles formed by the beam, wall, and building. The ratio of the sides of the triangles gives us \( \frac{y}{x} = \frac{y - 8}{27} \).

Solve the equation \( \frac{y}{x} = \frac{y - 8}{27} \) for y, substitute back into the expression for L, and find the minimum value of L using calculus by taking the derivative and setting it to zero.

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Key Concepts

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

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is essential for calculating the length of the beam, as it forms a right triangle with the wall and the distance from the wall to the building.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, we need to minimize the length of the beam, which can be modeled as a function of the height at which it touches the building. Understanding how to set up and solve optimization problems is crucial for finding the shortest beam.

Related Rates

Related rates are used in calculus to find the rate at which one quantity changes in relation to another. In this scenario, as the height of the beam changes, the length of the beam also changes. Understanding how to relate these rates will help in deriving the function that describes the beam's length in terms of its height.

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Related Practice

Textbook Question

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

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Applications

A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero.

Textbook Question

Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C

Textbook Question

Checking the Mean Value Theorem

Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.

f(x) = {sinx / x, −π ≤ x < 0

0, x = 0

Textbook Question

Absolute Extrema on Finite Closed Intervals

In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.

f(x) = x⁴ᐟ³, −1 ≤ x ≤ 8

Textbook Question

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫csc θ/(csc θ − sin θ) dθ