Question

The amount paid to a work crew varies jointly as the number of persons working and the length of time worked. If 5 workers earn \(5123.73\)dollars in 3.0 weeks, in how many weeks will 6 workers earn a total of \(6148.48\)dollars ?

Short answer

After calculating the constant of proportionality with the initial conditions, it takes approximately 3.6 weeks for 6 workers to earn $6148.48.

Step-by-step solution

Understanding Joint Variation

When a quantity varies jointly as two or more other quantities, it means the first quantity is directly proportional to the product of the other quantities. Here, the amount paid (A) varies jointly as the number of workers (n) and the time (t) which can be expressed as: A = k * n * t, where k is the constant of proportionality.

Finding the Constant of Proportionality

Using the given information, we can substitute A with \(5123.73, n with 5 workers, and t with 3.0 weeks to find the constant k: \)5123.73 = k * 5 * 3.0. Solve this equation to find the value of k.

Calculate the Constant k

To calculate k, divide \(5123.73 by the product of the number of workers and time: k = \)5123.73 / (5 * 3.0).

Setting up the Equation with New Values

With 6 workers to earn \(6148.48, we use the constant k to set up a new equation: \)6148.48 = k * 6 * t, where t is the unknown number of weeks we wish to solve for.

Solving for t

Now, solve for t by dividing \(6148.48 by the product of the constant k and 6 workers: t = \)6148.48 / (k * 6).

Finding the Number of Weeks

Substitute the value of k found in Step 3 into the equation to calculate t (the number of weeks).

Key concepts

Constant of Proportionality

The concept of the constant of proportionality is fundamental when dealing with joint variation. In these scenarios, one variable changes as the product of other variables change. The equation representing joint variation includes a special number called the constant of proportionality, denoted as 'k'. This number remains the same even as the other variables fluctuate.

In our exercise, we see an instance where the total earnings of a work crew is directly proportional to both the number of workers and the amount of time they work. By establishing that 5 workers earn $5123.73 in 3 weeks, we have enough information to calculate the constant of proportionality 'k', which represents a kind of unit rate of payment per worker per time period. Once we find 'k', we can use it to predict earnings under different conditions or with various numbers of workers and time frames.

Directly Proportional Relationships

Understanding directly proportional relationships is crucial in numerous mathematical and real-life applications. This type of relationship shows that as one quantity increases, another quantity increases at a constant rate. The term 'joint variation' often refers to a situation where one quantity is directly proportional to the product of two other quantities.

Using our textbook example, we see that the total pay is directly proportional to the number of workers and the time they work. This implies that doubling the number of workers, while keeping the time constant, would double the amount paid. Similarly, doubling the time worked, with the same number of workers, would also double the total earnings. Once the constant of proportionality is known, these calculations become straightforward.

Solving Algebraic Equations

Solving algebraic equations is a pivotal skill in algebra, where the goal is to find the value of unknown variables that make the equation true. The steps involved typically include simplifying expressions, isolating the variable of interest, and performing arithmetic operations in a methodic manner.

In the context of our work crew problem, after determining the constant of proportionality, we find ourselves with a new equation: \(6148.48 = k \times 6 \times t\). To find the unknown time 't', we need to isolate it by performing algebraic manipulations. Solving algebraic equations like this one often requires inversely applying the operations of multiplication or division, which furthers our understanding of the direct relationships among the variables involved.