Problem 22 - Precalculus with Calculus Previews

Question

The final length of a volcanic lava flow seems to decrease as the elevation of the lava vent from which it originates increases. An empirical study of Mt. Etna gives the final lava flow length \(L\) in terms of elevation \(h\) by the formula $$ L=23-0.0053 h $$

Step-by-Step Solution

Answer

Lava flow length decreases by 0.0053 km for every meter increase in elevation.

Step 1: Understand the Formula

The formula provided is \( L = 23 - 0.0053h \), where \( L \) is the final length of the lava flow in kilometers, and \( h \) is the elevation in meters from which the lava originates. The formula suggests that as the elevation \( h \) increases, the length \( L \) decreases.

Step 2: Identify the Variables

The primary variables in the equation are \( L \) (the length of the lava flow) and \( h \) (the elevation of the lava vent). Our goal is to determine how changes in \( h \) affect \( L \).

Step 3: Analyze the Coefficient

The coefficient of \( h \) is \(-0.0053\). This negative sign indicates an inverse relationship between \( h \) and \( L \); for every 1 meter increase in \( h \), \( L \) decreases by 0.0053 kilometers.

Step 4: Determine Specific Values

To find the exact length of a lava flow from a given elevation, substitute the value of \( h \) into the formula. For example, if \( h = 1000 \) meters, \( L = 23 - 0.0053 \times 1000 = 23 - 5.3 = 17.7 \) kilometers.

Step 5: General Interpretation

The formula shows that if the elevation is increased by 1000 meters, the lava flow's length decreases by about 5.3 kilometers, assuming all other factors remain constant.

Key Concepts

Inverse Relationship

An inverse relationship describes a situation where one variable increases while the other decreases. In the context of the lava flow formula, this means as the elevation (\( h \)) from which the lava originates increases, the final length of the lava flow (\( L \)) decreases. The formula provided is\[ L = 23 - 0.0053h \]Here, the coefficient \(-0.0053\) is negative, signifying this inverse relationship. To break this down further:

• As elevation \( h \) rises, the subtraction of the product \( 0.0053h \) from 23 increases, leading to a lesser final value of \( L \).
• The negative sign before \( 0.0053 \) is key; it suggests that there's a reversal in how these variables usually might interact (like how in a direct relationship they might both increase together).

This inverse relationship can help predict how changing one variable (elevation) impacts the other (lava flow length).

Empirical Formula

An empirical formula in this setting refers to a formula derived from observed data rather than theory. The lava flow equation \( L = 23 - 0.0053h \) is crafted from real-world measurements and observations of Mt. Etna’s lava flows. What this means is that scientists studied the pattern of lava flows over time and used their data to arrive at this formula:

• Such formulas provide practical, although sometimes simplified, models to predict outcomes.
• They may not account for every factor but are useful for capturing the primary relationship and trends.

The value here lies in its ability to offer reliable predictions based on actual phenomena, making it valuable for planning and research, even if it’s just one step in understanding complex geological events.

Variable Analysis

In variable analysis, we focus on understanding the roles and impacts of different variables within a formula. The formula for lava flow length involves two main variables:1. **Length of Lava Flow (\( L \))**: - Measured in kilometers. - Represents the final length the lava travels.2. **Elevation (\( h \))**: - Measured in meters. - Determines how high the lava originates from, significantly impacting \( L \).By conducting variable analysis, one can observe:

• Changes in the elevation \( h \) have a specified rate of impact on the flow length \( L \), as dictated by the coefficient \( -0.0053 \).
• Each variable plays a distinct role, with \( h \) serving as the controlling factor and \( L \) being the dependent outcome.

Such analysis helps to predict and assess how different conditions affect the phenomenon of lava flow, highlighting the critical interplay between elevation and flow length.