X is directly proportional to the square of Y. When X is 12, then Y is 2. Find the value of X when Y is 3.
27
Understanding Direct Proportionality between X and Y
The question states that X is directly proportional to the square of Y. This means that as the square of Y increases, X increases at a constant rate, and vice versa. We can express this relationship mathematically.
Mathematical Representation of Direct Proportionality
When X is directly proportional to the square of Y, we can write the relationship as:
\(X \propto Y^2\)
To turn this proportionality into an equation, we introduce a constant of proportionality, let's call it \(k\). The equation becomes:
\(X = kY^2\)
Here, \(k\) is a constant value that relates X and the square of Y.
Finding the Constant of Proportionality (k)
We are given that when X is 12, Y is 2. We can use these values to find the value of \(k\). Substitute X = 12 and Y = 2 into the equation \(X = kY^2\):
\(12 = k \times (2)^2\)
\(12 = k \times 4\)
Now, we can solve for \(k\):
\(k = \frac{12}{4}\)
\(k = 3\)
So, the constant of proportionality is 3. The specific relationship between X and Y is \(X = 3Y^2\).
Calculating X when Y is 3
Now that we know the relationship \(X = 3Y^2\), we can find the value of X when Y is 3. Substitute Y = 3 into the equation:
\(X = 3 \times (3)^2\)
\(X = 3 \times 9\)
\(X = 27\)
Therefore, when Y is 3, the value of X is 27.
Step-by-Step Solution Summary
- Identify the relationship: X is directly proportional to \(Y^2\).
- Write the equation: \(X = kY^2\).
- Use the first set of values (X=12, Y=2) to find \(k\): \(12 = k \times 2^2 \implies k=3\).
- Use the value of \(k\) (k=3) and the second value of Y (Y=3) to find X: \(X = 3 \times 3^2 = 3 \times 9 = 27\).
The value of X when Y is 3 is 27.
Revision Table: Direct Proportionality Key Concepts
| Concept | Description | Equation Form |
|---|---|---|
| Direct Proportionality | As one quantity increases, the other increases at a constant rate. Ratio is constant. | \(y \propto x\) or \(y = kx\) |
| Direct Proportionality to a Power | As one quantity increases, the other increases at a rate proportional to a power of the first. | \(y \propto x^n\) or \(y = kx^n\) |
| Constant of Proportionality (k) | The constant value relating the two proportional quantities. Found using one pair of values. | \(k = y/x\) or \(k = y/x^n\) |
Additional Information: Inverse and Joint Proportionality
Understanding proportionality is crucial in many mathematical and scientific contexts. Besides direct proportionality, two other common types are inverse proportionality and joint proportionality.
- Inverse Proportionality: \(y\) is inversely proportional to \(x\) means that as \(x\) increases, \(y\) decreases, such that their product is constant. The equation is \(y = k/x\) or \(xy = k\).
- Joint Proportionality: \(z\) is jointly proportional to \(x\) and \(y\) means that \(z\) is directly proportional to the product of \(x\) and \(y\). The equation is \(z = kxy\).
Being able to identify the type of proportionality and set up the correct equation with the constant \(k\) is the key to solving such problems.
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