Question

Write a general formula to describe the variation: \(d\) varies directly with \(t ; d=203\) when \(t=3.5\)

Short answer

The general formula is \[ d = 58t \].

Step-by-step solution

Understanding Direct Variation

When a variable d varies directly with another variable t, it means that d is equal to some constant k multiplied by t. Hence, we can express this relationship as: \[ d = k \times t \]

Substitute Given Values

Substitute the given values into the equation to find the constant k. We are given that d = 203 when t = 3.5. Thus, we substitute these values into the direct variation equation: \[ 203 = k \times 3.5 \]

Solve for k

To find the value of k, divide both sides of the equation by 3.5: \[ k = \frac{203}{3.5} \] Calculate the division: \[ k = 58 \]

Write the General Formula

Substitute the value of k back into the direct variation formula: \[ d = 58 \times t \] This is the general formula that describes the variation of d with t.

Key concepts

algebraic equations

Algebraic equations are mathematical statements that show the equality between two expressions. Equations can model real-world problems and relationships between variables. In our original exercise, we used an algebraic equation to express the direct variation relationship. We started with the general formula for direct variation: \[ d = k \times t \] where 'd' and 't' are variables and 'k' is the constant of variation. This equation tells us that 'd' depends directly on 't' and 'k'. Treat algebraic equations as tools that help you find unknown values based on given information. Applying algebraic principles, you can manipulate these equations to isolate and solve for specific variables. In our case, we isolated 'k' by dividing both sides of the equation by 't'. This process is fundamental in algebra and vital for solving real-life problems.

constant of variation

The constant of variation, often denoted by 'k', is a crucial component in direct variation equations. This constant represents the fixed ratio between two variables that vary directly with each other. In our example, we determined the constant of variation using the information given: \[ 203 = k \times 3.5 \] To find 'k', we divided 203 by 3.5, resulting in: \[ k = 58 \] This 'k' value means that for every unit increase in 't', 'd' increases by 58 units. The constant 'k' helps maintain the relationship between 'd' and 't' consistent, illustrating how they change together proportionally. Understanding the constant of variation is important because it reveals the rate at which one quantity changes relative to another. This concept is not only essential in mathematics but also in understanding relationships in science, economics, and engineering.

solving for k

Solving for 'k' is a critical step in direct variation problems. Here's how you can do it in a step-by-step manner:

• Start with the direct variation equation: \[ d = k \times t \]
• Substitute the known values into the equation. In our example, we used:\[ 203 = k \times 3.5 \]
• To find 'k', you need to isolate it on one side. Do this by dividing both sides by the corresponding variable:

\[ k = \frac{203}{3.5} \] Here, we divided 203 by 3.5, which equals 58. Solving for 'k' involves basic algebraic operations like division or multiplication. This step is essential as it provides the constant you will use in the general formula. After solving for 'k', the direct variation formula becomes:\[ d = 58 \times t \] Practicing these steps will help you quickly and accurately identify constants in other direct variation problems.