Problem 85 - Brief Applied Calculus

Question

A 5 -foot-long board is leaning against a wall so that it meets the wall at a point 4 feet above the floor. What is the slope of the board? [Hint: Draw a picture.]

Step-by-Step Solution

Answer

The slope of the board is \( \frac{4}{3} \).

Step 1: Understand the Problem

We have a right triangle formed by the board, the wall, and the floor. The board is the hypotenuse, the height from the floor to where the board meets the wall is 4 feet, and the length along the floor can be calculated. We need to find the slope of the board.

Step 2: Visualize the Triangle

The triangle's vertical side (opposite) is 4 feet, corresponding to the wall height. The hypotenuse is the board of length 5 feet. We need the base (adjacent) of the triangle to calculate the slope.

Step 3: Calculate the Base

Using the Pythagorean theorem, \[ c^2 = a^2 + b^2 \], where \( c = 5 \) and \( a = 4 \). Solve for \( b \) (the base): \[ 5^2 = 4^2 + b^2 \]. This simplifies to \( 25 = 16 + b^2 \) so \( b^2 = 9 \). Thus, \( b = 3 \) feet.

Step 4: Determine the Slope

The slope of the board is calculated as \( \frac{\text{opposite}}{\text{adjacent}} \) which is \( \frac{4}{3} \). This means the board rises 4 feet for every 3 feet it runs along the floor.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring 90 degrees. This triangle is often used in various fields of study, such as geometry and trigonometry. In our exercise, the right triangle is created by the board, the wall, and the floor. It includes:

• A right angle, where the wall and the floor meet.
• A vertical side, which goes from the floor to the point where the board meets the wall. In our case, this is 4 feet.
• A horizontal side, which spans along the floor. We calculated it to be 3 feet using the Pythagorean theorem.
• A hypotenuse, the board, which is 5 feet long.

This type of configuration frequently appears in slope calculation problems, making understanding the right triangle crucial.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It relates the lengths of the sides of a right triangle. In every right triangle, the relationship between the legs and the hypotenuse is given by: \[c^2 = a^2 + b^2 \]where:

• \(c\) is the hypotenuse, the longest side opposite the right angle
• \(a\) and \(b\) are the other two sides

In our scenario, the hypotenuse is the 5-foot board. The vertical leg (\(a\)) is 4 feet, while the horizontal leg (\(b\)) is what we need to find. Solving \(5^2 = 4^2 + b^2\), we find \(b = 3\). This shows how the Pythagorean theorem helps in calculating unknown sides in geometry.

Geometry

In geometry, understanding shapes and their properties helps solve various problems, like calculating slopes. Geometry involves points, lines, surfaces, and shapes. It's everywhere around us, from simple objects to complex buildings.
In the given problem, geometry helps us understand the relationship in the right triangle formed by the board, the wall, and the floor. By visualizing this triangle and understanding its properties, we can easily calculate the slope of the board.
This knowledge not only aids in academic problems but also in real-life situations where measurement and spatial understanding are crucial.

Hypotenuse

The hypotenuse is the longest side of a right triangle, always opposite the right angle. In problems involving slopes, such as ours, it plays a crucial role.

• It directly connects the endpoints of the vertical and horizontal sides.
• It's the board in this scenario, measuring 5 feet.
• It allows us to apply the Pythagorean theorem to find missing side lengths.

Knowing the length of the hypotenuse helps us understand the triangle's overall dimensions. Whether leaning boards or ladders, understanding the role of the hypotenuse aids in safety and design considerations.