Problem 17 - Algebra Form and Function

Question

In Exercises 14-17, assume the two quantities are directly proportional to each other. If \(s=35\) when \(t=25,\) find \(t\) when \(s\) is 14 .

Step-by-Step Solution

Answer

Answer: When the value of s is 14, the value of t is 10.

Step 1: Understand Direct Proportionality

Since the two quantities are directly proportional, we can say that: \(s = kt\) where \(k\) is the constant of proportionality.

Step 2: Using the given value\(s=35\) when \(t=25,\) find the constant of proportionality \(k\)

As we know that \(s = kt,\) substituting the given values of \(t\) and \(s\) we get: \(35 = k(25)\) Now, we can solve for \(k\): \(k = \frac{35}{25}\) \(k = 1.4\)

Step 3: Find the value of \(t\) when \(s = 14\)

Using the value of \(k\) we found in step 2, we write the equation for proportionality as: \(s = 1.4t\) Now, we can substitute the value of \(s\) provided in the problem: \(14 = 1.4t\) Next, we solve for \(t\): \(t = \frac{14}{1.4}\) \(t = 10\)

Step 4: Conclusion

When the value of \(s\) is 14, the value of \(t\) is 10.

Key Concepts

Constant of Proportionality

In the realm of proportional relationships, the constant of proportionality holds a key position. It is the fixed number that links two quantities in a relationship of direct proportionality. When two variables are directly proportional, the ratio between them remains constant. This means that as one variable changes, the other changes at a consistent rate determined by the constant. To determine the constant of proportionality in any problem, you can use the formula:

• If two quantities \( s \) and \( t \) are directly proportional, then \( s = kt \) where \( k \) is the constant.
• For example, if \( s = 35 \) when \( t = 25 \), \( k \) can be found by solving \( 35 = k \times 25 \). Here, \( k = \frac{35}{25} = 1.4 \).

By knowing \( k \), you can easily scale the relationship up or down, making it simple to find unknown values in proportional contexts.

Proportional Relationships

Proportional relationships are vital in understanding how quantities relate to each other in a linear fashion. When two variables are directly proportional, it means that they change at the same rate. Increasing one variable will proportionally increase the other, and this pattern holds true consistently. Key characteristics of proportional relationships include:

• A constant ratio exists between the two variables, as shown in the equation \( s = kt \).
• Graphically, a proportional relationship appears as a straight line passing through the origin on a graph of \( s \) versus \( t \).
• Real-world examples include speed and time in traveling, where distance is directly proportional to time if speed stays the same.

Understanding these relationships helps in predicting and solving problems where changes in one variable affect another directly.

Algebraic Equations

Algebraic equations form the backbone of solving real-life problems involving numbers. They allow us to represent relationships between quantities using variables and constants. In direct proportionality, algebraic equations help us quantify these relationships. Consider the equation \( s = kt \), which provides:

• A clear representation of how \( s \), a dependent variable, changes concerning \( t \), the independent variable.
• Using algebraic manipulations, one can easily isolate one variable to solve for it, such as finding \( t \) when \( s \) and \( k \) are known.
• For instance, if \( k = 1.4 \) and \( s = 14 \), substitute these values into the equation \( 14 = 1.4t \), manipulating it to find \( t = \frac{14}{1.4} = 10 \).

Mastering the use of algebraic equations and their manipulation is essential for tackling various mathematical problems effectively.