Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(s\) varies directly as \(t\). If \(t=10\), then \(s=18\).

Step-by-Step Solution

Answer

The formula is \( s = 1.8t \), and the constant \( k \) is 1.8.

Step 1: Understanding Direct Variation

In a direct variation problem, the relationship can be expressed using a formula. If \( s \) varies directly as \( t \), then the formula can be expressed as:\[s = kt\]where \( k \) is the constant of proportionality. This formula shows that \( s \) is directly proportional to \( t \).

Step 2: Substitute Known Values

The problem provides a specific condition to find the constant \( k \). We know that when \( t = 10 \), \( s = 18 \). Substitute these values into the equation \( s = kt \):\[18 = k \times 10\]

Step 3: Solve for the Constant of Proportionality

Now, solve the equation for \( k \). Divide both sides by 10 to isolate \( k \):\[k = \frac{18}{10} = 1.8\]So, the constant of proportionality \( k \) is 1.8.

Key Concepts

constant of proportionality

The concept of the constant of proportionality is fundamental to understanding direct variation. It is a constant value, denoted as \( k \), that defines the relationship between two directly proportional variables. When we say a variable \( s \) varies directly as \( t \), it means there is a constant multiplier that links these two variables. The formula representing this relationship is \( s = kt \). This means that as \( t \) changes, \( s \) will change in a consistent manner, determined by \( k \).
In our given problem, we were provided the specific condition that if \( t = 10 \), then \( s = 18 \). By substituting these values in the equation \( s = kt \), we calculate \( k \) as 1.8. Thus, for every unit increase in \( t \), \( s \) increases by 1.8 units.

linear equations

Linear equations are mathematical statements that describe a straight line when graphed. They are expressed in the standard form \( y = mx + b \) for most cases, but in direct variation, the relationship simplifies to \( y = kx \), where \( k \) is the constant of proportionality and there is no \( b \) term.
This is because, in direct variation, the line passes through the origin (0,0). Both \( s = kt \) and the typical linear form \( y = mx + b \) are similar in using a proportional constant to express how one quantity changes in relation to another. In our scenario, this translates to \( s = 1.8t \), which is a linear equation showing that \( s \) increases steadily as \( t \) increases. Since there is no additional constant (intercept), it begins at the origin and moves straight with slope \( k \).

proportional relationships

Proportional relationships represent scenarios where two quantities increase or decrease at the same rate. In such relationships, the ratio between the two quantities remains constant. This constant is often represented by the constant of proportionality, \( k \).
Direct variation is a key example of a proportional relationship. In the equation \( s = kt \), the relationship between \( s \) and \( t \) is direct and proportional. This means that for any two values \( t_1 \) and \( t_2 \), \( \frac{s_1}{t_1} = \frac{s_2}{t_2} \), holding \( k \) constant. When \( t \) doubles, so does \( s \), maintaining the same interval between them. Thus, such relationships are foundational in understanding how two variables can reliably predict each other, aided by the consistent proportionality constant.