Problem 21 - Algebra and Trigonometry

Question

13–22 ? Express the statement as an equation. Use the given information to find the constant of proportionality. \(s\) is inversely proportional to the square root of \(t .\) If \(s=100\) , then \(t=25 .\)

Step-by-Step Solution

Answer

The constant of proportionality \(k\) is 500.

Step 1: Understand Inverse Proportionality

The problem states that \(s\) is inversely proportional to the square root of \(t\). This means \(s\) is related to \(t\) such that as \(t\) increases, \(s\) decreases. The mathematical representation of this relationship is \(s = \frac{k}{\sqrt{t}}\), where \(k\) is the constant of proportionality.

Step 2: Substitute Known Values to Form an Equation

We are given that \(s = 100\) and \(t = 25\). Substitute these values into the equation \(s = \frac{k}{\sqrt{t}}\): \(100 = \frac{k}{\sqrt{25}}\).

Step 3: Simplify the Equation

Calculate the square root of 25, which is 5. Substitute this into the equation: \(100 = \frac{k}{5}\).

Step 4: Solve for the Constant of Proportionality

To solve for \(k\), multiply both sides of the equation by 5: \(100 \times 5 = k\). Thus, \(k = 500\).

Step 5: Write the Proportionality Equation

Now that we have found the constant of proportionality \(k\), the equation expressing \(s\) in terms of \(t\) is \(s = \frac{500}{\sqrt{t}}\).

Key Concepts

Constant of Proportionality

In the context of inverse proportionality, the constant of proportionality, often denoted as \( k \), is a crucial factor. This constant is what connects the variable \( s \) to the square root of \( t \), even as their values change. When \( s \) is inversely proportional to the square root of \( t \), this relationship is represented as \( s = \frac{k}{\sqrt{t}} \).

Finding the constant of proportionality involves substituting the known values from the problem into the equation and solving for \( k \). In our problem, we used \( s = 100 \) and \( t = 25 \) to find \( k = 500 \). This value of \( k \) remains constant regardless of the specific values of \( s \) and \( t \) used thereafter.

• It determines how the variables \( s \) and \( t \) are related.
• Once found, it allows us to predict \( s \) for any other value of \( t \).

Mathematical Representation

Mathematical representation is a systematic way of showing how variables relate to each other in a problem. In our exercise, we dealt with inverse proportionality, represented by the equation \( s = \frac{k}{\sqrt{t}} \).

Here's what this representation involves:

• \( s \) is the dependent variable which changes as \( t \) changes.
• \( k \) is the constant of proportionality, tying \( s \) and \( t \) together.
• \( \sqrt{t} \) is the square root of \( t \), showing that \( s \) depends inversely on the square root rather than \( t \) itself.

By writing \( s = \frac{500}{\sqrt{t}} \), we clearly express how \( s \) reduces as \( \sqrt{t} \) increases, reflecting the inverse relationship neatly in mathematical terms.

This mathematical formulation is key in converting word problems into solvable equations, allowing for clear deductions and predictions.

Proportionality Equation

The proportionality equation is a succinct way to express relationships between variables in mathematics. In inverse proportionality, the equation takes the form \( s = \frac{k}{\sqrt{t}} \), indicating that \( s \) decreases as \( t \) increases, when considering their relationship via the square root.

Understanding the construction of a proportionality equation aids in interpreting how variables interact:

• The numerator, \( k \), shows the intensity of the relation.
• The denominator, \( \sqrt{t} \), indicates how \( t \) affects \( s \).

In our problem, with \( k = 500 \), this equation becomes \( s = \frac{500}{\sqrt{t}} \). It implies a direct calculation method for \( s \) given any \( t \) and offers insight into behavior—predicting outcomes like decreases in \( s \) when \( t \) increases.

A proportionality equation serves as a powerful tool, allowing quick solutions once you identify the type of relationship, whether direct or inverse, between the relevant variables.