Problem 47 - Algebra 1

Question

You are climbing a 500 -foot cliff. By 1: 00 P.M. you have climbed 125 feet up the cliff. By 4: 00 P.M. you have reached a height of 290 feet. What is your rate of change in height?

Step-by-Step Solution

Answer

The rate of change in height is 55 feet/hour.

Step 1: Identify the heights and corresponding times

The climber starts at 125 feet at 1:00 p.m. and by 4:00 p.m., the climber has reached 290 feet. This gives two points (1,125) and (4,290) where the first component of the point represents the time in PM and the second component represents corresponding height in feet.

Step 2: Determine the change in height and time

The change in height delta(h) is the difference between the final height and the initial height. So, delta(h) = 290 - 125 = 165 feet. The change in time delta(t) is the difference between the final time and the initial time. So, delta(t) = 4 - 1 = 3 hours.

Step 3: Compute the rate of change

The rate of change is given by the formula: rate of change = delta(h) / delta(t) = 165 / 3 = 55 feet/hour.

Key Concepts

Slope Calculation

Slope calculation is a fundamental concept in understanding how things change over time or distance. It's like finding out how steep a hill is, or in this case, how fast you are climbing a cliff. The slope represents the rate of change, showing us how one variable changes in relation to another.

To calculate the slope, we typically use the formula:\[ \text{Slope} = \frac{\Delta y}{\Delta x} \]Here, \( \Delta y \) is the change in the vertical direction (like height), and \( \Delta x \) is the change in the horizontal direction (like time). In the cliff climbing exercise, the height changes from 125 feet to 290 feet, giving us a \( \Delta y \) of 165 feet. The time changes from 1:00 PM to 4:00 PM, giving us a \( \Delta x \) of 3 hours. So, the slope, or rate of change, is 55 feet/hour.

Knowing how to calculate slope helps us describe how quickly one thing is increasing or decreasing relative to another, whether it's climbing a cliff or observing any other variable changes in a straight-line manner.

Linear Equations

Linear equations are mathematical expressions that describe a straight-line relationship between two variables. These equations are super useful in analyzing scenarios where a constant rate of change is present, just like when climbing a cliff.

In a linear equation, the relationship between variables can be written in the form:\[ y = mx + b \]Here, \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) represents the slope (or rate of change), and \(b\) is the y-intercept, which is the starting point of the line when \(x = 0\). For our cliff climbing example, if we assume your climb started at 0 feet and time begins at 0, we could say 125 feet at 1:00 PM is the start, making it the y-intercept.

• **Slope (m)**: Represents how much you climb over each hour, calculated as 55 feet/hour in the exercise.
• **Intercept (b)**: Although not calculated in the exercise, it's useful to form the complete equation if starting from another point.

Using the slope and intercept can help predict future heights or analyze past climbs based on time.

Problem-Solving Skills

Developing strong problem-solving skills is essential when tackling exercises like the cliff climbing example. Breaking down problems into manageable steps helps you find solutions more effectively.

Here's how to improve these skills:

• **Understand the Problem**: Identify what you are solving. Here, the question was about finding the rate of change, which is essentially finding the slope.
• **Gather Information**: Note down what you know: initial height, final height, time, etc.
• **Apply Formulas**: Use the known formulas (like slope calculation) to solve the specific problem.
• **Check Your Work**: Always re-calculate and ensure that the solution makes sense in the context provided.

Applying these steps systematically will not only give accurate results in one problem but will also prepare you for various challenges involving rates of change and calculations. With practice, you'll become more adept at dissecting problems and identifying the best paths to solutions.