Problem 89 - 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: Understanding the Problem

The problem involves a right triangle formed by the floor, the wall, and the board as the hypotenuse. The board is 5 feet long (hypotenuse), and it meets the wall 4 feet above the floor (vertical side of the triangle).

Step 2: Setting Up the Equation for Slope

The slope of the board (line) is given by the change in vertical height divided by the change in horizontal distance. This means the slope is calculated as \( \frac{\text{vertical side}}{\text{horizontal side}} \).

Step 3: Using the Pythagorean Theorem

To find the horizontal side, apply the Pythagorean Theorem: \[ a^2 + 4^2 = 5^2 \]where \( a \) is the horizontal distance. Solve for \( a \): \[ a^2 + 16 = 25 \] \[ a^2 = 9 \] \[ a = 3 \].

Step 4: Calculating the Slope

Now that we know the horizontal distance \( a \) is 3, calculate the slope using:\[ \text{slope} = \frac{4}{3} \].

Step 5: Final Answer

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

Key Concepts

Pythagorean theorem

When dealing with right triangles, the Pythagorean theorem is a powerful tool. It relates the lengths of the sides of a right triangle. The formula is expressed as: \[ a^2 + b^2 = c^2 \] Here, \( a \) and \( b \) represent the lengths of the shorter sides, known as the legs, and \( c \) represents the hypotenuse, which is the longest side opposite the right angle. This theorem helps us find one side length when we know the other two. In our context, it allows us to calculate the horizontal side of the triangle (floor level) formed by the board leaning against the wall.

right triangle

A right triangle is a triangle with one angle measuring exactly 90 degrees. In this problem, the right triangle is formed by the wall, floor, and board. The wall acts as the vertical side of the triangle, the floor as the horizontal side, and the board as the hypotenuse. Right triangles are special because of their predictable properties, like the relationship between their sides, which is captured by the Pythagorean theorem. They also simplify the calculation of various aspects such as angles and slopes when solving geometric problems.

slope of a line

The slope of a line is a measure of its steepness or inclination. For any line represented on a two-dimensional plane, the slope is calculated as the ratio of the vertical change (rise) to the horizontal change (run). Mathematically, it is denoted by: \[ \text{slope} = \frac{\text{rise}}{\text{run}} \] For the board leaning against the wall, the rise is the vertical height of 4 feet (the height where the board touches the wall), and the run is the horizontal distance from the bottom of the wall to the point where the board touches the floor. This horizontal distance was calculated as 3 feet using the Pythagorean theorem. Therefore, the slope of the board is \( \frac{4}{3} \), indicating the board's angle of inclination relative to the floor. Understanding slope is crucial for analyzing the geometry of lines and their interaction within a space.