Direct Variation And Cube Roots Finding M In Terms Of N And Solving For N

Hey guys! Today, let's dive into a fun math problem involving direct variation and cube roots. We're given that m varies directly as twice the cube root of n. This might sound a bit complex at first, but we'll break it down step by step. Our main goal is to first find an expression for m in terms of n, and then determine the value of n when m equals 24. Ready to get started?

Understanding Direct Variation

Before we jump into the specifics, let's quickly recap what direct variation means. In simple terms, when two variables vary directly, it means that as one variable increases, the other variable increases proportionally, and vice versa. Mathematically, we can express this relationship as y = kx, where y and x are the variables, and k is the constant of variation. This constant k is super important because it tells us the exact relationship between the variables. So, if we double x, y will also double, and if we halve x, y will also be halved. This consistent proportionality is the heart of direct variation.

In our problem, m varies directly as twice the cube root of n. This means we can write the relationship as m = k(2∛n), where k is the constant of variation we need to find. The cube root part adds a little twist, but the fundamental principle of direct variation remains the same. To solve this, we'll first need to find the value of k using the initial conditions given to us, which are m = 8 when n = 27. Once we have k, we'll have a complete expression for m in terms of n. Then, we can use that expression to find the value of n when m = 24. So, let's move on and see how we can find this constant of variation. It's like finding the secret ingredient that connects m and n!

Finding the Constant of Variation (k)

Alright, let's roll up our sleeves and find the constant of variation, k. This is a crucial step because once we have k, we'll have the key to unlock the relationship between m and n. We're given that m = 8 when n = 27. Remember our equation from before: m = k(2∛n). To find k, we'll simply substitute these values into the equation and solve for it. So, let's plug in m = 8 and n = 27:

8 = k(2∛27)

Now, we need to figure out the cube root of 27. Think of it this way: what number, when multiplied by itself three times, gives you 27? That's right, it's 3! So, ∛27 = 3. Let's substitute that into our equation:

8 = k(2 * 3)

This simplifies to:

8 = k(6)

Now, to isolate k, we'll divide both sides of the equation by 6:

k = 8 / 6

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

k = 4 / 3

There we have it! The constant of variation, k, is 4/3. This means that for every change in 2∛n, m changes by 4/3 times that amount. Now that we've found k, we can write the complete expression for m in terms of n. This expression will allow us to predict the value of m for any given value of n, and vice versa. So, let's move on to the next step and write out this expression. We're one step closer to solving the whole problem!

Expressing m in Terms of n

Okay, with the constant of variation k = 4/3 in our hands, we can now write the expression for m in terms of n. This is like writing the formula that connects these two variables. Remember our general equation: m = k(2∛n). All we need to do is substitute the value of k we just found into this equation. So, here we go:

m = (4/3)(2∛n)

We can simplify this a little bit by multiplying the constants together:

m = (8/3)∛n

And there it is! This is the expression for m in terms of n. It tells us exactly how m changes as n changes. The cube root of n is multiplied by 8/3 to give us the value of m. This equation is super useful because now we can plug in any value of n and find the corresponding value of m, or vice versa. It's like having a magic formula that relates these two variables. This is the answer to the first part of our problem – we've successfully expressed m in terms of n.

Now that we have this expression, we can move on to the second part of the problem, which asks us to find the value of n when m = 24. This will involve a little bit of algebraic manipulation, but we've already done the hard part. We have the equation, and we have the value of m, so it's just a matter of plugging it in and solving for n. So, let's get to it!

Finding the Value of n When m = 24

Alright, guys, let's tackle the final part of the problem: finding the value of n when m = 24. We've already done the groundwork by finding the expression for m in terms of n, which is m = (8/3)∛n. Now, we'll use this expression to solve for n when m = 24. This is like having a map and knowing your destination; now we just need to follow the route to get there.

First, let's substitute m = 24 into our equation:

24 = (8/3)∛n

Now, we need to isolate the cube root of n. To do this, we'll multiply both sides of the equation by the reciprocal of 8/3, which is 3/8:

24 * (3/8) = ∛n

Let's simplify the left side of the equation. 24 multiplied by 3/8 is:

(24 * 3) / 8 = 72 / 8 = 9

So, our equation now looks like this:

9 = ∛n

To get rid of the cube root, we need to cube both sides of the equation. This means raising both sides to the power of 3:

9³ = (∛n)³

9 cubed (9 * 9 * 9) is 729, and cubing the cube root of n simply gives us n:

729 = n

And there we have it! The value of n when m = 24 is 729. This is the answer to the second part of our problem. We've successfully used our expression to find the value of one variable given the other. This shows the power of direct variation and how we can use it to solve real-world problems.

Conclusion

Great job, everyone! We've successfully navigated through this problem involving direct variation and cube roots. We started by understanding the concept of direct variation, then we found the constant of variation, k. With k in hand, we expressed m in terms of n as m = (8/3)∛n. Finally, we used this expression to find the value of n when m = 24, which turned out to be 729. This problem demonstrates how we can use mathematical relationships to solve for unknown variables and understand how quantities are related to each other.

Remember, guys, the key to solving these kinds of problems is to break them down into smaller, manageable steps. First, understand the relationship between the variables, then find the constant of variation, and finally, use the expression to solve for the unknowns. With practice, these problems will become much easier, and you'll be able to tackle even more complex mathematical challenges. Keep up the great work, and I'll see you in the next math adventure!

• Direct Variation
• Cube Root
• Constant of Variation
• Expression for m in terms of n
• Finding the value of n