Question

If \(y\) is directly proportional to the cube of \(x,\) and \(y\) is 27.2 when \(x\) is \(11.4,\) find \(y\) when \(x\) is 24.9

Short answer

First, find the constant of proportionality \(k\) using the given values (\(y=27.2\) when \(x=11.4\)), then use \(k\) to find \(y\) when \(x=24.9\).

Step-by-step solution

Understand Proportionality

Since we know that \(y\) is directly proportional to the cube of \(x\), we can say that \(y = kx^3\), where \(k\) is the constant of proportionality.

Solve for the Constant of Proportionality

To find the constant \(k\), use the given values where \(y\) is 27.2 when \(x\) is 11.4. Substitute these values into the equation to get \(27.2 = k(11.4)^3\). Solve for \(k\) by dividing both sides by \((11.4)^3\).

Calculate the Value of \(k\)

Carrying out the calculation from step 2, \(k = \frac{27.2}{(11.4)^3}\). Use a calculator to find the value of \(k\).

Find \(y\) When \(x\) is 24.9

With the value of \(k\) found, substitute \(x = 24.9\) into the equation \(y = kx^3\) to find the new value of \(y\).

Calculate the New Value of \(y\)

Use the previously calculated value of \(k\) and multiply it by \((24.9)^3\) to get the new value of \(y\). This will give us the value of \(y\) when \(x\) is 24.9.

Key concepts

Proportionality Constant

In mathematics, the proportionality constant is a vital concept, especially when dealing with direct proportions. Imagine a scenario where one variable changes exactly in tune with another - this is where the proportionality constant, often denoted as 'k', plays its part.

Consider two variables, let's say 'y' and 'x'. If 'y' varies directly as a power of 'x', we describe this as a proportional relationship. In the given exercise, 'y' is said to be directly proportional to the cube of 'x'. This means 'y' changes as the cube of 'x' changes. The relationship can be written as \(y = kx^3\), where 'k' symbolizes our proportionality constant. This constant remains, well, constant, and it's the key to maintaining the proportional relationship no matter how 'x' and 'y' may vary.

The value of 'k' can be determined by using known values of 'x' and 'y'. Once 'k' is found, it paves the way for solving any further problems involving these variables. It's like having the secret ingredient to a perfect recipe; with 'k', you can predict and calculate other pairs of 'x' and 'y' that will preserve their direct proportionality.

Proportional Relationships

Proportional relationships are all about a consistent ratio between variables. When one variable moves, the other follows suit in a predictable pattern. It's the harmony of mathematics! Put simply, if we have two variables, 'A' and 'B', and A is in direct proportion to B, any change in A leads to a proportional change in B and vice versa.

In the context of the exercise, 'y' alters as the cube of 'x' does, creating an equation that looks like \(y = kx^3\). This cubic relationship means that whenever 'x' is scaled up or down, 'y' will respond by getting cube times larger or smaller accordingly. It's a bit like inflating or deflating a balloon - as the air volume (our 'x') changes, the size of the balloon ('y') expands or contracts based on the same pattern; in this case, the cube of the change in volume.

The beauty of proportional relationships is that they provide a clear and direct link between variables, making them invaluable for predictions and solving real-world problems. These relationships allow us to develop models that can simulate various situations.

Solving Proportionality Problems

Tackling proportionality problems can initially seem daunting, but with the right approach, it becomes a cinch. In our exercise, for instance, the equation \(y = kx^3\) sets the stage. Once 'k' is known, the rest is a straightforward plug-and-play. This means that if you're given the value of one variable and you need to find the other, it's only a matter of substituting the known values into the equation and solving.

Here’s how it works: Start with identifying what you already know, and then use this to find your proportionality constant, 'k'. This step requires careful substitution of given values and performing the arithmetic accurately. Once you’ve got 'k', it becomes your magic key! Any future values of 'x' can be used to find the corresponding 'y' by simply putting 'x' back into the equation.

To cement the concept, let's see a quick bullet list of the process:

• Understand the type of proportionality (direct, inverse, etc.) involved.
• Determine the proportionality constant using known variable values.
• Substitute the new variable value into the equation to find its counterpart.
• Verify the solution, ensuring it holds true to the nature of the proportionality.

By methodically following these steps, solving proportionality problems becomes just another solvable puzzle, awaiting your keen problem-solving skills.