Question

A ramp for wheelchair access to the gym has a slope of 5\(\%\) ( that is, the ramp rises 5 feet vertically for every 100 feet of horizontal distance). The entire ramp is built on level ground, and the entrance to the gym is 2 feet above the ground. What is the horizontal distance, in feet, between the ends of the ramp? A. 4 B. 10 C. 40 D. 100 E. 400

Short answer

Answer: C. 40 feet

Step-by-step solution

Understand the meaning of a 5% slope

A 5% slope means that for every 100 feet of horizontal distance, the ramp rises 5 feet vertically. We can express this relationship as a fraction: \(\frac{5}{100}\), which represents the rise (vertical distance) over the run (horizontal distance).

Set up the proportion using the given vertical height

We know that the entrance to the gym is 2 feet above the ground, and this is the total vertical distance (rise) of the ramp. To find the horizontal distance (run), we can set up a proportion involving the known slope and the vertical height: \(\frac{5}{100} = \frac{2}{x}\) Here, x represents the horizontal distance we need to find.

Cross-multiply and solve for x

To solve for the horizontal distance x, cross-multiply the fractions: \(5 * x = 100 * 2\) This gives us: \(5x = 200\) Now, divide both sides by 5: \(x = 40\)

Choose the correct answer and state the conclusion

Thus, the horizontal distance between the ends of the ramp is 40 feet, which corresponds to option C. So, the correct answer is: C. 40

Key concepts

Proportion in Math

Understanding proportion in math is crucial for solving many real-world problems, including slope percentage problems like the one in our exercise. Proportion is about the relationship between two ratios or fractions being equal to each other. In simpler terms, it's about maintaining the same ratio or balance when comparing quantities.

For instance, in our ramp problem, if a ramp rises 5 feet for every 100 feet of horizontal distance, that's a ratio which can be written as a fraction: \(\frac{5}{100}\). When the ramp's height is different, like 2 feet in the gym entrance, we still want to keep the same rise-to-run ratio. This is where we use a proportion: \(\frac{5}{100} = \frac{2}{x}\), where \(x\) is the unknown horizontal distance we are trying to find. Proportions show up everywhere in daily life, such as in recipes, maps, models, and even in financial transactions, such as currency conversion.

Cross-Multiplication

Cross-multiplication is a handy tool for solving proportions. It allows you to find the missing value in a proportion by multiplying diagonally across the equal sign. It simplifies the process of solving for the unknown variable.

Here's how it works: take the proportion \(\frac{a}{b} = \frac{c}{d}\) with \(d\) being unknown. Cross-multiply by multiplying \(a\) with \(d\) and \(c\) with \(b\), then set the products equal to each other to get \(a \times d = b \times c\). Isolate \(d\) by dividing both sides by \(a\). This gives you \(d = \frac{b \times c}{a}\).

In our gym ramp example, this method swiftly leads us to the solution: \(5 \times x = 100 \times 2\), which quickly simplifies to \(x = 40\) once we divide by 5. Using cross-multiplication makes the solving process more efficient and reduces the chance of errors, especially when dealing with more complex proportions.

Geometry in Real Life Application

Geometry is not just a subject in school; it's a practical tool we use in everyday life, often without even realizing it. In the context of our exercise, the geometry of designing a ramp involves understanding and calculating slopes. It's a direct application of geometric principles to ensure accessibility, safety, and functionality in buildings.

The slope of a ramp is a geometric concept that affects how steep the ramp is. Inaccessible or unsafe slopes can make ramps unusable for people in wheelchairs. Thus, calculating the correct slope percentage, as we did in our example, is not only a mathematical exercise but also a critical component of inclusive design. By understanding how to apply geometric principles in these contexts, we learn to create better, more thoughtful designs in our communities.