- Define the relationship as \(y^2 = k \cdot x^3\).
- Calculate the constant \(k\) using the given point \((4, 2)\), yielding \(k = \frac{1}{16}\).
- Form the final equation by substituting \(k\): \(y^2 = \frac{1}{16} x^3\).
- The correct choice is \(\boxed{y^2 = \frac{1}{16} x^3}\).
Explanation
- Understanding the Relationship
The problem states that the square of \(y\) varies directly as the cube of \(x\). In mathematical terms, "varies directly" implies a relationship of the form \(y^2 = k \cdot x^3\), where \(k\) is a constant of variation. We are given that when \(x = 4\), \(y = 2\). We can use this information to determine the value of the constant \(k\).
- Determining the Constant of Variation
Substitute \(x = 4\) and \(y = 2\) into the equation \(y^2 = k \cdot x^3\):
\[2^2 = k \cdot (4)^3\]
\[4 = k \cdot 64\]
Now, solve for \(k\):
\[k = \frac{4}{64} = \frac{1}{16}\]
Substituting the value of \(k\) back into the general equation, we get:
\[y^2 = \frac{1}{16} x^3\]
This matches the first option provided in the problem.
- Verification and Conclusion
To ensure that the other options are incorrect, we can test them with a different point on the curve defined by \(y^2 = \frac{1}{16} x^3\). Let \(x = 8\). Then:
\[y^2 = \frac{1}{16} \cdot 8^3 = \frac{512}{16} = 32\]
\[y = \sqrt{32} = 4\sqrt{2} \approx 5.657\]
Checking this point \((8, 4\sqrt{2})\) against the other options:
- \(y = \frac{1}{2}x \implies 4\sqrt{2}
eq \frac{1}{2} \cdot 8 = 4\)
- \(xy = 8 \implies 8 \cdot 4\sqrt{2} = 32\sqrt{2}
eq 8\)
- \(x^3 y^2 = 128 \implies 8^3 \cdot 32 = 512 \cdot 32 = 16384
eq 128\)
Since only the first equation holds for the given proportionality, it is the correct representation.
Examples
Direct variation is a fundamental concept used in physics and engineering to model relationships between physical quantities. For example, in fluid dynamics, the drag force experienced by an object moving through a fluid often varies directly as the square of its velocity. If you measure the force at a specific speed, you can determine the proportionality constant and use it to predict the force at any other speed, which is crucial for vehicle safety and design efficiency.