Verified Answer :
- Determine the constant of proportionality \(k\) for each problem using the given data pairs (\(k = y/x\)).
- Apply the formula \(y = kx\) (or equivalent) to solve for the target variables in each scenario.
- Part 1: \(y = 3 \times 25 = 75\).
- Part 2: \(C = 2.1 \times 28 = 58.8\).
- Part 3: \(75 = 3x \Rightarrow x = 25\).
- Final answers: \(\boxed{75}\), \(\boxed{\$58.80}\), \(\boxed{25}\).
Explanation
- Analysis of Direct Variation Problems
The concept of direct variation (or direct proportionality) describes a relationship between two variables, say \(x\) and \(y\), such that one is a constant multiple of the other. Mathematically, this is expressed as \(y = kx\), where \(k\) is the constant of proportionality. To solve for missing values, we first determine the constant \(k\) using a known pair of values, then apply that \(k\) to the new situation. Here we have three separate problems to analyze:
- Find \(y\) given \(y\) varies directly as \(x\), \((x, y) = (12, 36)\), find \(y\) for \(x = 25\).
- Find the cost of 28 books given the cost of 15 books is \(\$31.50\).
- Find \(x\) given \(y\) is directly proportional to \(x\), \((x, y) = (6, 18)\), find \(x\) for \(y = 75\).
- Solving the Three Problems
For the first problem, we use the direct variation formula \(y = kx\). With \(y = 36\) and \(x = 12\), the constant of proportionality is \(k = \frac{36}{12} = 3\). To find \(y\) when \(x = 25\), we calculate \(y = 3 \times 25 = 75\).
For the second problem, we define \(C\) as the cost and \(n\) as the number of books, so \(C = kn\). Given 15 books cost \(\$31.50\), the constant is \(k = \frac{31.50}{15} = 2.1\). The cost of 28 books is thus \(C = 2.1 \times 28 = 58.8\). Expressed in currency, the cost is \(\$58.80\).
For the third problem, \(y = kx\) with \(y = 18\) and \(x = 6\). The constant is \(k = \frac{18}{6} = 3\). To find \(x\) when \(y = 75\), we solve \(75 = 3x\), which gives \(x = \frac{75}{3} = 25\).
- Conclusion and Results
The calculated values for the three problems are as follows:
- When \(y\) varies directly as \(x\), for \(x = 25\), \(y = \boxed{75}\).
- The cost of 28 exercise books is \(\boxed{\$58.80}\).
- When \(y\) is directly proportional to \(x\), for \(y = 75\), \(x = \boxed{25}\).
Examples
Direct variation is a fundamental tool for scaling in everyday life. For instance, if you are painting a wall, the amount of paint required is directly proportional to the area of the wall. If you know that 2 liters of paint cover 10 square meters, you can calculate that 1 liter covers 5 square meters (\(k = 0.2\)). If you then need to paint a 50-square-meter room, you use the relationship \(P = 0.2 \times A\), so \(P = 0.2 \times 50 = 10\) liters. This concept is essential for everything from cooking recipes and budget planning to construction material estimation.