Problem 95 - College Algebra with Modeling and Visualization

Question

Thickness of Runways Heavier aircraft require runways with thicker pavement for landings and takeoffs. A pavement 6 inches thick can accommodate an aircraft weighing \(80,000\) pounds, whereas a 12 -inch-thick pavement is necessary for a \(350,000\) -pound plane. The relation between pavement thickness \(t\) in inches and gross weight \(W\) in thousands of pounds can be modeled by \(W=C a^{t} .\) (a) Find values for \(C\) and \(a\) (b) How heavy an airplane can a 9 -inch-thick runway accommodate? (c) What is the minimum thickness for a \(242,000\) -pound plane?

Step-by-Step Solution

Answer

(a) Solve for \( C \) and \( a \). (b) Use equation for 9-inch. (c) Solve equation for given weight.

Step 1: Understand the relationship

The relationship between pavement thickness \( t \) and gross weight \( W \) is modeled by the equation \( W = C a^t \). We need to find the constants \( C \) and \( a \) using the given conditions.

Step 2: Set up the equations

Using the points provided: for \( t = 6 \), \( W = 80 \), thus the equation is \( 80 = C a^6 \); for \( t = 12 \), \( W = 350 \), thus the equation is \( 350 = C a^{12} \). Note that \( W \) is given in thousands of pounds.

Step 3: Solve for \( C \) and \( a \)

Divide the two equations: \( \frac{350}{80} = \frac{Ca^{12}}{Ca^6} \). This simplifies to \( \frac{350}{80} = a^6 \). Calculate \( a = \left(\frac{350}{80}\right)^{1/6} \). Use \( a \) value to find \( C \) by substituting back into \( 80 = C a^6 \).

Step 4: Calculate \( W \) for the 9-inch pavement

Use the derived \( C \) and \( a \) to calculate \( W \) when \( t = 9 \): \( W = C a^9 \). Substitute the values of \( C \) and \( a \) to find \( W \).

Step 5: Determine minimum thickness for 242,000-pound plane

Use the equation \( 242 = C a^t \) since 242,000 pounds convert to \( W = 242 \) thousand pounds. Solve for \( t \) using known \( C \) and \( a \): \( t = \frac{\log(242/C)}{\log(a)} \). Calculate \( t \).

Key Concepts

Mathematical Modeling

Mathematical modeling is like creating a bridge between abstract mathematics and real-world problems. In this exercise, we use a mathematical model to understand the relationship between the thickness of a runway and the weight of an aircraft. The model provides a formula, specifically \( W = Ca^t \), where \( W \) is the aircraft's weight in thousands of pounds, \( t \) is the thickness of the pavement, and \( C \) and \( a \) are constants that need to be determined based on real-world data.

• **Real-World Data**: We start by using known scenarios. For example, a 6-inch thick pavement can support 80,000 pounds, and a 12-inch thick pavement can support 350,000 pounds.
• **Translating to Mathematics**: This data is then translated into equations: \( 80 = Ca^6 \) and \( 350 = Ca^{12} \).

Mathematical modeling helps break down complex situations into manageable equations, making it easier to analyze real-world scenarios.

Exponential Functions

Exponential functions describe situations where quantities grow or shrink at rates proportional to their current value. In our runway problem, the thickness of the pavement impacts the weight it can support exponentially. The relationship is expressed as \( W = Ca^t \), which is an example of an exponential function with base \( a \).

• **Properties**: Exponential functions have unique properties like rapid growth or decay. Here, as the thickness increases, the weight capacity grows exponentially.
• **Solving**: To find \( a \), we divide the given equations to get \( \frac{350}{80} = a^6 \), and solve \( a = \left( \frac{350}{80} \right)^{1/6} \). This step is crucial as it uncovers the fundamental growth rate of our function.

This demonstrates how exponential functions can model growth patterns effectively, connecting the mathematics to physical interpretations.

Problem-Solving

Problem-solving in algebra often involves breaking down the problem into smaller, manageable parts using logical reasoning. For the aircraft and runway exercise, the steps are straightforward yet require meticulous calculations and logical deductions.

• **Identify Unknowns**: Start by recognizing what needs to be found, such as the constants \( C \) and \( a \), the weight capacity for a new thickness, or the minimum thickness for a given weight.
• **Use Known Information**: We use the relationships developed earlier, like \( W = Ca^t \), and plug in known values to determine unknown variables.
• **Logical Deduction**: To find the minimum thickness for a 242,000-pound plane, set \( W = 242 \), translate into the equation \( 242 = Ca^t \), and solve for \( t \) using logarithmic transformations: \( t = \frac{\log(242/C)}{\log(a)} \).

Effective problem-solving requires understanding the mathematical relationships, applying algebraic techniques, and using logical reasoning to achieve the right solution.