Problem 59 - Thomas Calculus

Question

The variable \(s\) is proportional to \(t,\) and \(s=25\) when \(t=75 .\) Determine \(t\) when \(s=60\)

Step-by-Step Solution

Answer

\(t = 180\)

Step 1: Understand the Proportional Relationship

Since the variable \(s\) is proportional to \(t\), this relationship can be expressed as \(s = k \cdot t\), where \(k\) is the constant of proportionality. The goal is to find \(k\) using the given values.

Step 2: Calculate the Constant of Proportionality \(k\)

We are given that \(s = 25\) when \(t = 75\). Using the formula \(s = k \cdot t\), substitute the values: \(25 = k \cdot 75\). Solve for \(k\) by dividing both sides by 75: \[ k = \frac{25}{75} = \frac{1}{3} \]

Step 3: Determine \(t\) When \(s=60\)

Now, we have the constant \(k = \frac{1}{3}\). Use the proportionality equation \(s = k \cdot t\), and substitute \(s = 60\) and \(k\): \[ 60 = \frac{1}{3} \cdot t \]To find \(t\), multiply both sides by 3:\[ t = 60 \cdot 3 = 180 \]

Key Concepts

Constant of Proportionality

In a proportional relationship, one quantity changes in direct proportion to another. This can be represented mathematically as \( s = k \cdot t \), where \( s \) and \( t \) are variables, and \( k \) is the constant of proportionality. This constant, \( k \), is a fixed number that shows the rate at which the variables change in relation to each other.
To determine \( k \), you use known values of \( s \) and \( t \). In the example exercise, we knew that \( s = 25 \) when \( t = 75 \).
This allowed us to set up and solve the equation:

• \( 25 = k \cdot 75 \)

By solving this, we found \( k = \frac{1}{3} \). The constant \( k \) helps us predict other values of \( s \) or \( t \) when given one of them.

Solving Equations

Solving equations is a fundamental part of working with direct proportionality. In our exercise, once we have determined the constant of proportionality, we can use it to find unknown values of the related variables. To find the unknown variable, rearrange the proportionality equation accordingly.
For example, once we knew that \( k = \frac{1}{3} \), and we wanted to determine \( t \) for \( s = 60 \), we first wrote the equation:

• \( 60 = \frac{1}{3} \cdot t \)

Then, we solved for \( t \) by isolating the variable. This required us to multiply both sides by the reciprocal of \( \frac{1}{3} \) (which is 3) to clear the fraction.

• \( t = 60 \cdot 3 = 180 \)

This step-by-step approach helps in clearly seeing how each piece of information fits into the overall problem, leading to logical and grounded solutions.

Mathematical Modeling

Mathematical modeling is the process of using mathematical structures and concepts to represent real-world scenarios. In our problem, the relationship between \( s \) and \( t \) shows how mathematical models help in predicting unknown variables when there is a proportional relationship.
With the equation \( s = k \cdot t \) serving as our model, we can input known data to find unknown values. This is useful in many real-life situations where measurements or predictions need to be made based on known data.

• For instance, in economics, such a model can represent cost vs. production levels.
• In physics, it might represent speed over time for constant acceleration.

The ability to model situations using mathematics makes it possible to derive logical solutions and make informed decisions based on the analyses.