Problem 67 - Understanding Elementary Algebra with Geometry

Question

An escalator in a mall carries customers a vertical distance of 22 feet while traveling a horizontal distance of 58 feet. Find the grade of the escalator.

Step-by-Step Solution

Answer

37.93%

Step 1: Identify the right triangle

Visualize the situation as a right triangle where the vertical distance is one leg and the horizontal distance is the other leg. The escalator’s path will be the hypotenuse.

Step 2: Write the grade formula

The grade of an escalator is typically given as the ratio of the vertical distance to the horizontal distance. Mathematically, this is expressed as \(\text{Grade} = \frac{\text{Vertical Distance}}{\text{Horizontal Distance}}\).

Step 3: Substitute given values

Insert the given vertical and horizontal distances into the formula. So it becomes \(\text{Grade} = \frac{22}{58}\).

Step 4: Simplify the fraction

Simplify the fraction \(\frac{22}{58}\) if possible. Both 22 and 58 can be divided by their greatest common divisor, 2, giving \(\frac{11}{29}\).

Step 5: Convert to percentage

To express the grade as a percentage, you multiply the simplified fraction \( \frac{11}{29} \) by 100. This gives approximately \( \frac{11}{29} \times 100 \approx 37.93\text{%}\).

Key Concepts

right triangle problem

A right triangle is a triangle in which one of the angles is exactly 90 degrees. In this problem, we can visualize the escalator as forming a right triangle with the horizontal floor of the mall and an imaginary line from the top end of the escalator, perpendicular to the floor. The vertical distance (22 feet up) represents one leg of the right triangle, while the horizontal distance (58 feet forward) represents the other leg. The escalator itself is the hypotenuse, the longest side of the right triangle. Understanding that this forms a right triangle helps us in calculating various properties, such as the grade or slope of the escalator.

grade calculation

The grade of an escalator tells us how steep it is. It's calculated as the ratio of the vertical rise to the horizontal run. This is mathematically written as \( \text{Grade} = \frac{\text{Vertical Distance}}{\text{Horizontal Distance}} \). For our problem, the vertical distance is 22 feet, and the horizontal distance is 58 feet. Therefore, the grade formula becomes \( \text{Grade} = \frac{22}{58} \). This ratio gives a fraction which indicates how many feet the escalator rises for each foot it runs horizontally. Higher grades mean steeper escalators, which are harder to climb.

simplifying fractions

Simplifying fractions involves reducing them to their simplest form. The fraction \( \frac{22}{58} \) can be simplified by dividing both the numerator (22) and the denominator (58) by their greatest common divisor. The greatest common divisor of 22 and 58 is 2. So when we divide both 22 and 58 by 2, we get the simplified fraction \( \frac{11}{29} \). Simplifying fractions makes them easier to understand and use in further calculations.

expressing ratios as percentages

To express a ratio as a percentage, we convert the fraction into a percentage. This involves multiplying the simplified fraction by 100. For our case, the fraction \( \frac{11}{29} \) becomes approximately 37.93% when multiplied by 100. This means the escalator rises 37.93 feet for every 100 horizontal feet, which helps to better visualize the steepness of the escalator. Converting ratios to percentages makes it easier to compare different grades or slopes, as percentages are a more intuitive measure for most people.