Problem 20 - Precalculus Mathematics for Calculus

Question

Express the statement as an equation. Use the given information to find the constant of proportionality. \(P\) is directly proportional to \(T .\) If \(T=300,\) then \(P=20\).

Step-by-Step Solution

Answer

The constant of proportionality \( k \) is \( \frac{1}{15} \).

Step 1: Understand Direct Proportionality

When a quantity \( P \) is directly proportional to another quantity \( T \), it means that \( P = kT \), where \( k \) is a constant of proportionality.

Step 2: Write the Proportionality Equation

According to the problem statement, \( P \) is directly proportional to \( T \). Therefore, the equation for direct proportionality is: \( P = kT \).

Step 3: Substitute Given Values

We are given that when \( T = 300 \), \( P = 20 \). Substitute these values into the equation: \( 20 = k \times 300 \).

Step 4: Solve for the Constant of Proportionality

Solve the equation \( 20 = 300k \) for \( k \). Divide both sides by 300 to isolate \( k \): \( k = \frac{20}{300} = \frac{1}{15} \).

Key Concepts

Constant of Proportionality

When two quantities are directly proportional, a constant of proportionality defines this relationship. It's a specific number, often symbolized by \( k \), that connects one variable to another in a direct way. In our problem, the relationship between \( P \) and \( T \) is such that \( P = kT \). This means you can find \( P \) by multiplying \( T \) by \( k \). Once \( T \) and \( P \) values are given, you can determine \( k \) using these steps:
\( \)

• Substitute the values of \( T \) and \( P \) into \( P = kT \).
• Solve for \( k \) by isolating it on one side of the equation.

After working through the example, we found that the constant of proportionality \( k \) is \( \frac{1}{15} \). Knowing \( k \) helps predict \( P \) for any \( T \) value by simply multiplying \( \frac{1}{15} \) by the given \( T \). This forms the backbone of directly proportional relationships.

Proportionality Equation

A proportionality equation shows how two quantities are related through a constant multiplication factor. For direct proportionality, if \( P \) is proportional to \( T \), we express it with the equation \( P = kT \).
This equation defines:

• \( P \) - the dependent variable.
• \( T \) - the independent variable.
• \( k \) - the constant of proportionality.

The beauty of this formula is its simplicity. Once \( k \) is known, predicting \( P \) for any \( T \) is straightforward. For example, doubling \( T \) will double \( P \) since \( k \) stays the same. The proportionality equation is a powerful tool used in various fields like physics, economics, and beyond.

Substitution Method

The substitution method involves replacing variables with given values to solve for unknowns, particularly useful in proportionality equations. Here’s how it works:
1. Start with a proportionality equation such as \( P = kT \).
2. Substitute known values of \( P \) and \( T \) into the equation.
3. Solve the equation for the unknown, often the constant \( k \).
This strategy simplifies direct proportionality problems by breaking them down into manageable steps. For instance, by substituting \( P = 20 \) and \( T = 300 \) into \( P = kT \), we solve \( 20 = 300k \) to find \( k = \frac{1}{15} \). Thus, substitution is a vital method for calculating unknowns and ensuring the equation holds true for specific conditions.