Given y is inversely proportional to √x, and x = 36 when y = 36. What is the value of x when y = 54?
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Solving Inverse Proportion Problems
This problem involves the concept of inverse proportionality. When one quantity is inversely proportional to another (or its root), their product (or the product of one quantity and the root of the other) is a constant.
The question states that y is inversely proportional to \(\sqrt{x}\). Mathematically, this relationship can be written as:
\(y \propto \frac{1}{\sqrt{x}}\)
To turn this proportionality into an equation, we introduce a constant of proportionality, let's call it \(k\). So, the equation is:
\(y = \frac{k}{\sqrt{x}}\)
Finding the Constant of Proportionality
We are given that when \(x = 36\), \(y = 36\). We can use these values to find the value of \(k\).
Substitute \(x = 36\) and \(y = 36\) into the equation:
\(36 = \frac{k}{\sqrt{36}}\)
Calculate the square root of 36:
\(\sqrt{36} = 6\)
So the equation becomes:
\(36 = \frac{k}{6}\)
To find \(k\), multiply both sides by 6:
\(k = 36 \times 6\)
\(k = 216\)
Now we have the specific relationship between \(y\) and \(x\):
\(y = \frac{216}{\sqrt{x}}\)
Calculating x when y = 54
We need to find the value of \(x\) when \(y = 54\). Use the relationship equation we just found:
\(y = \frac{216}{\sqrt{x}}\)
Substitute \(y = 54\) into the equation:
\(54 = \frac{216}{\sqrt{x}}\)
To solve for \(\sqrt{x}\), we can rearrange the equation:
\(\sqrt{x} = \frac{216}{54}\)
Perform the division:
\(\sqrt{x} = 4\)
To find \(x\), we need to square both sides of the equation:
\((\sqrt{x})^2 = 4^2\)
\(x = 16\)
So, the value of \(x\) when \(y = 54\) is 16.
Summary of Inverse Proportion Calculation
Here is a summary of the steps taken:
- Identify the inverse proportionality relationship: \(y \propto \frac{1}{\sqrt{x}}\) which means \(y = \frac{k}{\sqrt{x}}\).
- Use the initial values (\(x=36, y=36\)) to calculate the constant \(k\).
- Write the specific equation relating \(y\) and \(x\) using the found \(k\).
- Use the new value (\(y=54\)) in the equation to solve for \(x\).
| Step | Calculation | Result |
|---|---|---|
| Find k | \(36 = \frac{k}{\sqrt{36}} \implies k = 36 \times 6\) | \(k = 216\) |
| Find x when y=54 | \(54 = \frac{216}{\sqrt{x}} \implies \sqrt{x} = \frac{216}{54} = 4\) | \(x = 4^2 = 16\) |
Revision Table: Inverse Proportion Concepts
| Concept | Description | Mathematical Form |
|---|---|---|
| Direct Proportion | As one quantity increases, the other increases proportionally. | \(y \propto x\) or \(y = kx\) |
| Inverse Proportion | As one quantity increases, the other decreases proportionally. | \(y \propto \frac{1}{x}\) or \(y = \frac{k}{x}\) |
| Inverse Proportion to Root | As one quantity increases, the other decreases proportionally to the root. | \(y \propto \frac{1}{\sqrt{x}}\) or \(y = \frac{k}{\sqrt{x}}\) |
| Constant of Proportionality (k) | A constant value relating two quantities in a proportion. | \(k = \frac{y}{x}\) (Direct) or \(k = yx\) (Inverse) or \(k = y\sqrt{x}\) (Inverse to √x) |
Additional Information: Understanding Proportionality
Proportionality describes how two quantities change in relation to each other. There are two main types:
- Direct Proportionality: Two quantities, \(A\) and \(B\), are directly proportional if their ratio is constant. This means if \(A\) doubles, \(B\) also doubles. If \(A\) is halved, \(B\) is also halved. The relationship is \(A = kB\), where \(k\) is the constant of proportionality.
- Inverse Proportionality: Two quantities, \(A\) and \(B\), are inversely proportional if their product is constant. This means if \(A\) doubles, \(B\) is halved. If \(A\) is halved, \(B\) doubles. The relationship is \(A = \frac{k}{B}\), where \(k\) is the constant of proportionality.
In this specific problem, the inverse relationship is not with \(x\) directly, but with its square root, \(\sqrt{x}\). This is why the relationship is written as \(y = \frac{k}{\sqrt{x}}\).
Problems involving proportionality usually require you to first use a given pair of values to find the constant of proportionality (\(k\)), and then use that constant with a new value of one variable to find the corresponding value of the other variable.
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