P & Q: Unveiling Proportionality And Inverse Relationships

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Hey math enthusiasts! Let's dive into a fun problem involving proportionality and inverse relationships. This isn't your average math class; we're going to break down how to find a formula for p in terms of q. We'll use the given information to unveil the relationship and simplify it to the core. So, buckle up; we're about to explore how proportionality and inverse relationships work in the world of variables! Ready to unravel the secrets of p and q? Let's get started!

Unpacking the Proportionality of P and W

Alright, first things first, let's look at the heart of our problem: p is directly proportional to the square root of w. This means that as the square root of w increases, p increases proportionally. It's like they're dancing together, always in sync. We can express this relationship mathematically as:

p ∝ √w

To turn this proportionality into an equation, we need to introduce a constant of proportionality, which we'll call k. So, our equation becomes:

p = k√w

Now, the problem gives us a crucial piece of information: p = 56 when w = 49. We can use this to find the value of k. Plugging these values into our equation, we get:

56 = k√49

Since the square root of 49 is 7, the equation simplifies to:

56 = 7k

To find k, we divide both sides by 7:

k = 56 / 7

k = 8

So, our equation now becomes:

p = 8√w

We've successfully established a direct relationship between p and w. But, our ultimate goal is to find a formula for p in terms of q. Don't worry, we are just one step away from our final result. This is a crucial step in understanding how variables are related. This foundation will enable us to tackle the next phase of our problem.

Transition to Inverse Proportionality

Next up, we need to know the inverse relationship between w and q. Get ready to reverse the direction; instead of increasing together, one goes up while the other goes down. Here's where the problem shows that w is inversely proportional to the cube of q. Mathematically, it's expressed as:

w ∝ 1/q³

This means that as q increases, w decreases, and vice versa. Again, we introduce a constant of proportionality, let's call it m, to change this into an equation:

w = m/ q³

We're given that w = 2 when q = 2. Let's use this info to solve for m. Substitute the values:

2 = m / 2³

Since 2³ = 8:

2 = m / 8

Multiply both sides by 8 to solve for m:

m = 2 * 8

m = 16

So, our equation becomes:

w = 16 / q³

We now have equations for both the direct and inverse relationships. We have everything we need to find the direct relationship between p and q.

Combining the Relationships to Find P in Terms of Q

We're getting closer to our final goal. We have two core equations:

  1. p = 8√w
  2. w = 16 / q³

Our mission is to find p in terms of q. We can do this by substituting the value of w from the second equation into the first equation. Basically, we're replacing w with its equivalent expression in terms of q. Let's perform the substitution:

p = 8√(16 / q³)

Now, let's simplify this. We can take the square root of the numerator and the denominator separately:

p = 8(√16 / √q³)

Since the square root of 16 is 4:

p = 8(4 / √q³)

p = 32 / √q³

This is the expression for p in terms of q. However, to present this formula in its simplest form, we should rewrite the denominator to make sure it looks neat and is easy to understand. We can rewrite √q³ as q^(3/2), which means q raised to the power of 3/2:

p = 32 / q^(3/2)

To make it even simpler, we can bring q^(3/2) up to the numerator by changing the sign of the exponent:

p = 32q^(-3/2)

So, the formula for p in terms of q in its simplest form is p = 32/q^(3/2) or p = 32 * q^(-3/2). That's the answer, folks! We've successfully navigated through the world of proportionality and inverse relationships to arrive at our solution! The final result clearly expresses the direct relationship between p and q. Now we can easily predict how p will change according to any given value of q. The formula gives us a clear understanding of the mathematical relationship between the variables, and how they change together. You guys did great!

Key Takeaways

To recap what we've learned, here are the key steps:

  • Understand Proportionality: Recognize that p ∝ √w means p increases as the square root of w increases. Use k as the constant of proportionality.
  • Solve for the Constant: Using the given values (p = 56 when w = 49), find the value of k to get the equation p = 8√w.
  • Understand Inverse Proportionality: Recognize that w ∝ 1/q³ means w decreases as q³ increases. Use m as the constant of proportionality.
  • Solve for the Second Constant: Using the given values (w = 2 when q = 2), find the value of m to get the equation w = 16/q³.
  • Substitution: Substitute the expression for w (in terms of q) into the equation for p. That gives us p = 8√(16 / q³).
  • Simplify: Simplify the equation, and rewrite it in the simplest form, p = 32 / q^(3/2) or p = 32 * q^(-3/2).

Great job on getting through the problem! You've learned how to dissect proportionality and inverse relationships, solve for constants, and manipulate equations. Understanding how variables relate to each other through proportionality and inverse proportionality is a fundamental skill in math and science. Keep practicing and exploring, and you'll become math masters in no time! Keep up the great work and happy calculating!