Question

A road rises 4 feet over the course of 100 horizontal feet. What is the slope of a line representing the road? A. 4 B. 0.04 C. 2.5 D. 25

Short answer

B. 0.04

Step-by-step solution

- Define Slope

The slope of a line is defined as the ratio of the rise (vertical change) to the run (horizontal change). Mathematically, it is expressed as: \[ \text{slope} = \frac{\text{rise}}{\text{run}} \]

- Identify Rise and Run

In this problem, the rise is given as 4 feet and the run is given as 100 feet.

- Calculate Slope

Using the formula for slope, substitute the given values: \[ \text{slope} = \frac{4 \text{ feet}}{100 \text{ feet}} = 0.04 \]

- Choose the Correct Answer

Among the given choices (A: 4, B: 0.04, C: 2.5, D: 25), the correct slope is 0.04, which corresponds to option B.

Key concepts

mathematical reasoning

Understanding mathematical reasoning is key to solving many problems in geometry and algebra. When we talk about the slope of a line, we use logical steps to identify and relate the elements involved. First, you need to understand the formula for the slope, \( \frac{𝑟𝑖𝑠𝑒}{𝑟𝑢𝑛} \). By applying this formula to given values like in the exercise, we can find the answer systematically. Mathematical reasoning helps you align each step logically, making sure that every calculation leads you closer to the solution. This skill will not only help you in this specific problem but also in understanding more complex mathematical concepts that follow similar logical patterns.
Examples could be:

• Solving equations
• Understanding proofs
• Performing exact calculations

Reasoning clearly and correctly ensures that all your mathematical work stands on a strong foundation.

ratio

A ratio is a way to compare two quantities. In the given problem, you compare the rise (vertical distance) to the run (horizontal distance) to determine the slope. The ratio \( \frac{4 \text{ feet}}{100 \text{ feet}} \) simplifies to 0.04, which describes the slope of the road. Understanding ratios is not just about solving this problem but is also useful in various real-life applications such as:

• Cooking recipes where you need to understand ingredient proportions
• Financial calculations involving interests and investments
• Understanding maps and scales

Simplifying ratios helps make complex problems easier to handle by breaking them down into more understandable parts. For instance, the ratio 4:100 can be simplified to 1:25, providing a clear, simplified view of the relationship.

geometry

Geometry involves studying shapes, sizes, and properties of figures and spaces. When calculating the slope of a line, you are working within this branch of mathematics. The slope describes how steep a line is, which is essential in understanding various geometric shapes and figures. Geometric concepts you may encounter and utilize include:

• Lines and angles
• Triangles and their properties
• Coordinate geometry, where you plot points and lines

In our exercise, identifying the rise and run as a vertical and horizontal change corresponds to geometric thinking. By interpreting these distances on a coordinate plane, you gain further insights into how lines behave and interact. This understanding will be helpful as you progress to more complex geometric problems like those involving curves, slopes of other shapes, and even three-dimensional figures.