Question

You are supervising the construction of a roller coaster for young children. For the first 20 feet of horizontal distance, the track must rise off the ground at a constant rate. After your crew has constructed 5 feet of horizontal distance, the track is 1 foot off the ground. During construction, the park passes a regulation that the track for any roller coaster for young children must not be higher than 5 feet off the ground. Will your roller coaster pass inspection when it is completed?

Short answer

Yes, the roller coaster will pass the inspection. After 20 feet of constructed track, it will be 4 feet off the ground, which is less than the maximum allowed height of 5 feet as per the new regulations.

Step-by-step solution

- Determine the Rate of Rise

Given the rise in height for 5 feet of horizontal distance is 1 foot, we can conclude that for each foot of horizontal distance, the track rises \( \frac{1}{5} \) feet. This is the constant rate of rise.

- Calculate the Height After 20 Feet

With 20 feet of horizontal distance, and a rate of rise of \( \frac{1}{5} \) feet per foot of horizontal distance, we get a total height of \( 20 * \frac{1}{5} = 4 \) feet.

- Compare with the Regulation Height

The calculated height of 4 feet is less than the maximum allowed height of 5 feet for children's roller coasters according to the new regulations.

Key concepts

Linear Relationships

Imagine you're creating a drawing with a ruler; you're making a straight line. That's what a linear relationship looks like in mathematics. It's a special connection between two quantities that increase or decrease at a constant rate. In our roller coaster scenario, the construction of the track represents a linear relationship between the horizontal distance and the height of the track. As the track gets longer, it also gets higher, but it does it in a way that's predictable and consistent—just like a straight line on a graph.

Think of it this way: for each foot of horizontal track added, if the height consistently goes up by the same amount, the relationship between the track's length and height is linear. It allows us to make easy predictions. For instance, knowing that after 5 feet the track is 1 foot high, we can predict the height at any point along the first 20 feet without actually building the track. This predictability is what makes understanding linear relationships so powerful in various fields, from engineering to finance.

Algebraic Expressions

Algebraic expressions are like recipes that help us make mathematical dishes. They use numbers, variables (stand-ins for numbers), and operations like addition and multiplication to describe relationships between different things. In our roller coaster example, the rate of rise can be expressed as an algebraic expression. Here, the expression is \( \frac{1}{5} \), which tells us for every unit of horizontal distance, the rise is \( \frac{1}{5} \) of a foot.

Using this expression, we can cook up the height of the roller coaster at any point—without physically measuring it! Algebrically, if 'x' is the horizontal distance, then the height 'h' can be expressed as \( h = \frac{1}{5}x \). This simple expression succinctly communicates the constant rise of the track, irrespective of how long the track is. It's like a magic formula that uncovers the secret height at any point you wish to know on the roller coaster.

Slope of a Line

The slope of a line is essentially a way to measure the steepness or incline of that line. Think of it as counting the number of stairs you'd climb or descend on a hillside—this number would give an idea of the slope. Mathematically, the slope is the ratio of the change in height (rise) to the change in horizontal distance (run). When we look at the roller coaster, the slope represents how quickly or slowly the roller coaster track rises as it extends horizontally.

In the given exercise, the slope is calculated as the rise over run, which is \( \frac{1}{5} \). This means for every 5 units of horizontal distance, the track will rise by 1 unit, indicating a gentle slope - perfect for young children's roller coasters. Since a line with a constant slope is straight and doesn't curve or bend, it makes our job to check against safety regulations a cakewalk. After building 20 feet, the track being only 4 feet off the ground proves that the slope is kept at a child-friendly grade, and the roller coaster project successfully meets the safety standards set by the park regulations.