Verified Answer :
- The constant of variation \(k\) for the first problem is \(\frac{36}{12} = 3\), leading to \(y = 3 \times 25 = \boxed{75}\).
- The constant of variation \(k\) for the second problem is \(\frac{31.50}{15} = 2.1\), leading to a cost of \(2.1 \times 28 = \boxed{\$58.80}\).
- The constant of variation \(k\) for the third problem is \(\frac{18}{6} = 3\), leading to \(x = \frac{75}{3} = \boxed{25}\).
Explanation
- Problem Analysis and Objective
We are analyzing three separate problems involving direct proportionality. A relationship is directly proportional if it can be represented by the equation \(y = kx\), where \(k\) is the constant of variation. To solve these, we determine \(k\) using the given pair of values, and then apply it to find the unknown quantity.
- For the first problem: \(y\) varies directly as \(x\), with \(y = 36\) when \(x = 12\).
- For the second problem: The cost \(C\) of \(n\) forks is directly proportional, with \(C = \$31.50\) for \(n = 15\).
- For the third problem: \(y\) is directly proportional to \(x\), with \(y = 18\) when \(x = 6\).
We will solve each step-by-step.
- Solving the First Proportionality Problem
To find the value of \(y\) when \(x = 25\), given \(y = 36\) when \(x = 12\):
First, find the constant of variation \(k = \frac{y}{x} = \frac{36}{12} = 3\).
Now, substitute \(k\) and the new value of \(x\) into the equation:
\(y = 3 \times 25 = 75\).
Thus, the value of \(y\) is \(75\).
- Solving the Second Proportionality Problem
To find the cost of \(28\) forks given that \(15\) forks cost \(\$31.50\):
First, find the constant of variation \(k = \frac{C}{n} = \frac{31.50}{15} = 2.1\).
The cost per fork is \(\$2.10\). Now, find the cost of \(28\) forks:
\(C = 2.1 \times 28 = 58.8\).
Thus, the cost of \(28\) forks is \(\$58.80\).
- Solving the Third Proportionality Problem
To find the value of \(x\) when \(y = 75\), given that \(y = 18\) when \(x = 6\):
First, find the constant of variation \(k = \frac{y}{x} = \frac{18}{6} = 3\).
Now, substitute \(k\) and the new value of \(y\) into the equation \(y = kx\):
\(75 = 3 \times x \implies x = \frac{75}{3} = 25\).
Thus, the value of \(x\) is \(25\).
- Conclusion and Final Answers
In summary, the values calculated are:
- When \(x=25\), \(y = 75\).
- The cost of \(28\) forks is \(\$58.80\).
- When \(y=75\), \(x = 25\).
The final answers are \(\boxed{75}\), \(\boxed{\$58.80}\), and \(\boxed{25}\) respectively.
Examples
Direct proportionality is a fundamental concept in daily life, such as when calculating the total cost of items. For example, if you know the price of one kilogram of apples, you can calculate the total cost for any number of kilograms you wish to buy by using the formula \(C = k \cdot w\), where \(C\) is the total cost, \(w\) is the weight, and \(k\) is the price per kilogram. This allows for efficient budgeting and comparison shopping whenever you are at the grocery store.