Problem 72 - Calculus with Concepts in Calculus

Question

A bookcase is 7 feet tall, 4 feet wide, and 1 foot deep. What is the minimum vertical distance that is necessary in order to raise the bookcase from a horizontal position on the floor to a vertical position against the wall?

Step-by-Step Solution

Answer

The minimum vertical distance to raise the bookcase is \( \sqrt{50} \) feet.

Step 1: Understand the Problem

We have a bookcase that initially lies flat on the floor, and we want to stand it up against the wall. The bookcase dimensions are 7 feet tall, 4 feet wide, and 1 foot deep.

Step 2: Identify Maximum Height at Edge of Rotation

When standing the bookcase up, one of its edges will act as a pivot point on the ground. The challenge is to determine the vertical height raised from the floor when the bookcase is at a 90-degree angle, touching the wall.

Step 3: Determine the Hypotenuse with Pythagoras

Consider the base (4 feet wide) and the vertical height from pivot to the opposite bottom corner of the bookcase when rotated. Using the Pythagorean theorem, we calculate: hypotenuse = \( \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} \).

Step 4: Calculate Vertical Distance

The vertical distance (rise in height) when pivoting around the edge touching the floor is the hypotenuse calculated earlier. The vertical height lifted from the ground is \( \sqrt{50} \) feet.

Key Concepts

Geometry and Spatial Understanding

Geometry is an essential part of understanding the physical world around us. It deals with the properties and relations of points, lines, surfaces, and solids. When approaching a geometry problem, like the bookcase, it's vital to break it down into shapes and dimensions we know. In this exercise, the bookcase is a rectangular prism.
This shape consistency helps us apply geometric principles such as the Pythagorean Theorem, especially since we're dealing with right triangles formed by the transitions of the bookcase.

• The bookcase lying flat on the ground can be thought of as a right triangle when pivoted.
• The dimensions of the bookcase act as the edges of this right triangle.

Grasping these geometric relations allows us to visualize the problem better and find solutions that relate physical dimensions and abstract mathematical theories.

Problem Solving with the Pythagorean Theorem

Effective problem-solving often involves breaking complex problems into smaller, more manageable parts. With our bookcase, one of the key tools is the Pythagorean Theorem, which applies in any right-angled triangle. This theorem states that the square of the length of the hypotenuse (the longest side) is the sum of the squares of the other two sides.

For the bookcase, as it rotates, it forms a right triangle from the width (4 feet), the depth (1 foot), and the hypotenuse which is the distance from the pivoting point (corner on the ground) to its opposite corner that is being raised to vertical.

• Identify the two sides or 'legs' of the triangle: 7 feet and 1 foot.
• Apply: Pythagorean Theorem: \[ c = \sqrt{a^2 + b^2} \]
• Substitute dimensions for 'a' and 'b': \[ c = \sqrt{7^2 + 1^2} \]

This results in finding the hypotenuse, which is the minimum vertical distance the bookcase must rise to stand upright. Understanding the problem incrementally, using geometric and algebraic knowledge, can make problem-solving more efficient and effective.

Mathematical Modeling for Real-Life Application

Mathematical modeling is a powerful tool that helps translate real-life problems into mathematical forms that can be solved systematically. In the bookcase problem, this means representing the situation with geometric shapes and mathematical equations. Such models can predict real outcomes and assist in decision-making.

• Mathematical modeling in this case starts by recognizing the bookcase as a geometric shape that changes orientation.
• By modeling the bookcase's movement through the Pythagorean Theorem, we calculate precise distances needed for its real-world movement: standing it up vertically.
• This model not only gives a theoretical answer but can be directly applied to predict and plan how much space is needed to safely and efficiently maneuver the bookcase in actual practice.

Thus, mathematical modeling not only aids in solving textbook problems but also provides solutions that are applicable to everyday logistical challenges.