Question

If \(y\) is inversely proportional to the square of \(x,\) and \(y\) has the value 1.55 when \(x\) is \(7.38,\) find \(y\) when \(x\) is 44.2.

Short answer

Using the constant \(k\), found to be \(k = 1.55 \times (7.38)^2\), calculate the value of \(y\) with the given \(x\) value, resulting in \(y = \frac{k}{(44.2)^2}\).

Step-by-step solution

Understanding Inverse Proportionality

If one value is inversely proportional to the square of another, we can write this as an equation: \(y = \frac{k}{x^2}\), where \(k\) is the constant of proportionality.

Finding the Constant of Proportionality

Using the given values, we can plug them into the equation to solve for \(k\): \(1.55 = \frac{k}{7.38^2}\). Solving for \(k\) gives us: \(k = 1.55 \times 7.38^2\).

Calculate the Value of k

Calculate the value of \(k\) by squaring 7.38 and multiplying by 1.55: \(k = 1.55 \times (7.38)^2\).

Determine y for x = 44.2

Now, we use the value of \(k\) to find \(y\) when \(x\) is 44.2. Plug these values into the formula: \(y = \frac{k}{(44.2)^2}\).

Final Calculation

Solve the equation for \(y\) with the known \(k\), to find the value of \(y\) when \(x = 44.2\).

Key concepts

Proportional Relationships

In the world of mathematics, proportional relationships are a cornerstone concept. They occur when two quantities maintain a constant ratio to each other. Imagine you're baking cookies and for every cup of flour you add, you need half a cup of sugar. As you multiply the amount of flour, the amount of sugar increases proportionally. In mathematical terms, if we say the sugar (\( y \) is directly proportional to flour (\( x \)), the relationship can be expressed as \( y = k \times x \) where 'k' is a consistent number known as the constant of proportionality.

But what if we're talking about inverse proportionality, like in our original exercise? This is like saying, as the circle of your pizza gets larger, the crust gets thinner. The formula flips, and we describe it using \( y = \frac{k}{x} \) or in the case of our exercise, \( y = \frac{k}{x^2} \) because the relationship is proportional to the square of 'x'. Understanding these relationships is crucial to solving many real-world problems.

Constant of Proportionality

The constant of proportionality is what keeps the relationship between two inversely proportional value steady. Think of it as the invisible glue that holds our mathematical relationship together. For inverse proportionality, we can represent the constant of proportionality with 'k' in the equation \( y = \frac{k}{x^2} \), with 'k' remaining unchanged regardless of the values of 'y' and 'x'.

How do we find 'k'? We use known values of 'y' and 'x', just like in the solution step 2. Once we've established what 'k' is, it's like finding a secret decoder ring—we can unlock the value of 'y' for any 'x' we encounter in the future. It's a snapshot of the invariant relationship between 'y' and 'x', and why a consistent 'k' is so critical for proportional equations.

Solving Equations

Solving equations is a fundamental skill in math. It's all about isolating the variable we're curious about, step by step. When handling proportional equations, we usually pick a known pair of values for 'x' and 'y', and use them to find our constant 'k'. Once 'k' is in hand, it's substitution time! We can plug in the new value of 'x' and use the same 'k' to find the corresponding 'y', just as done in steps 4 and 5.

Here's a tip: when you're working through the steps, keep an eye on the equation's operations — are we squaring, taking a square root, multiplying, dividing? Doing operations in the correct order is key to a correct solution. It's all about following these methodical steps to reveal the mysteries hidden within the numbers.