Question

If \(y\) is directly proportional to the square of \(x,\) and \(y\) is 5570 when \(x\) is 172 find \(y\) when \(x\) is 382

Short answer

When \( x \) is 382, \( y \) is approximately 27502.

Step-by-step solution

Understand the relationship

Directly proportional means that as one amount increases, another amount increases at a constant rate. In mathematical terms, if y is directly proportional to the square of x, we can write this relationship as: \( y = kx^2 \), where \( k \) is the constant of proportionality.

Find the constant of proportionality

Using the information given, when \( x = 172 \), \( y = 5570 \). Substitute these values into the equation to find \( k \): \( 5570 = k(172)^2 \). Solve for \( k \) by diving both sides by \( (172)^2 \): \( k = \frac{5570}{172^2} \).

Calculate the value of the constant

Perform the division to find the constant \( k \): \( k = \frac{5570}{29584} \approx 0.1883 \). This value is approximate and could be represented with greater precision depending on the context.

Use the constant to find y when x is 382

Now that we have the constant \( k \), we use the equation \( y = kx^2 \) to find \( y \) when \( x = 382 \). Substitute \( k \) and \( x \) into the equation: \( y = 0.1883(382)^2 \).

Solve for y

Calculate the value of \( y \) by squaring 382 and multiplying by \( k \): \( y = 0.1883 \times 145924 \approx 27502.3472 \). The value of \( y \) is approximately 27502 when \( x \) is 382.

Key concepts

Constant of Proportionality

Understanding the constant of proportionality is essential in exploring direct proportionality in mathematics. It's the 'k' in our equation that firmly holds two varying quantities together. Think of it as the glue that maintains their relationship at a consistent and predictable rate, no matter how much one quantity changes. In our exercise, we identified the constant by using given values of the variables to create an equation, \( y = kx^2 \), and then we unraveled it to find \( k \).When we solved for \( k \), what we essentially did was to pin down the exact scale of their relationship: we now know exactly how much \( y \) will respond to changes in \( x \). This constant is the heart of understanding and predicting outcomes within directly proportional scenarios. Once determined, it's a powerful tool, enabling us to forecast the value of \( y \) for any given \( x \), without additional complex calculations.

Proportional Relationships

A proportional relationship between two quantities means that they increase or decrease in unison. If you think of it like synchronized swimming, both participants (the quantities in our case) are performing a perfectly coordinated routine (growing or shrinking in value). In a direct proportion, like we see with \( y \) being directly proportional to the square of \( x \), we witness this dance where \( y \) will always be the square of \( x \), scaled by the constant of proportionality.

Recognizing Proportional Relationships

It's important to recognize this type of relationship as it simplifies our understanding of how two quantities relate. We can depict this visually with straight-line graphs passing through the origin or algebraically as simple equations. In both instances, the proportionality constant \( k \) is our focus point, acting as the slope or gradient of our graphical line, or the multiplier in our algebraic equation. By keeping an eye on \( k \) and knowing its value, solving problems becomes a matter of substitution and straightforward computation.

Mathematical Problem-Solving

Critical mathematical problem-solving skills come into play when we approach an exercise like finding \( y \) when \( x \) is 382. We first understand the scenario and establish the direct proportionality. Next, we identify what we know and what needs to be discovered. The constant of proportionality \( k \) unlocks the ability to correlate \( y \) and \( x \) values effectively.

Applying Systematic Steps

Our approach is systematic: define the relationship, deduce \( k \) using known quantities, and then apply \( k \) to find unknowns. Problem-solving in mathematics often follows this pattern of exploration, deduction, and application. It's about breaking the problem down into manageable pieces and then logically piecing them back together to form a complete picture of the solution.The steps taken in calculating the value of \( y \) when \( x \) is 382 are a demonstration of mathematical reasoning at work. We've taken abstract concepts and used them to not just solve a problem, but also to predict and understand how variables in direct proportion behave in different circumstances.