Okay, let's see. I need to find the value of x when y is 3, given that y varies inversely as the square root of x. Hmm, inverse variation. So, inverse variation means that as one quantity increases, the other decreases proportionally. The formula for inverse variation is y = k/x, where k is the constant of variation. But wait, in this case, it's y varies inversely as the square root of x. So, I think the formula should be y = k / sqrt(x). Let me confirm that. Yes, if y varies inversely with sqrt(x), then y = k / sqrt(x). 

Alright, so the first step is to find the constant k. They give us that when x is 24, y is 15. So I can plug those values into the equation to find k. Let me write that down:

15 = k / sqrt(24)

So, to solve for k, I need to multiply both sides by sqrt(24). Let's compute sqrt(24). Well, sqrt(24) is the same as sqrt(4*6) which is 2*sqrt(6). So sqrt(24) = 2*sqrt(6). Therefore, k = 15 * 2 * sqrt(6) = 30*sqrt(6). Let me check that again. Yes, because 15 times 2 is 30, and 30 times sqrt(6) is 30*sqrt(6). So k is 30*sqrt(6).

Now that we have k, the equation relating y and x is y = 30*sqrt(6) / sqrt(x). The question asks for x when y is 3. So we can set up the equation:

3 = 30*sqrt(6) / sqrt(x)

Now, solve for sqrt(x). Let's rearrange the equation. Multiply both sides by sqrt(x):

3*sqrt(x) = 30*sqrt(6)

Then divide both sides by 3:

sqrt(x) = (30*sqrt(6)) / 3

Simplify 30 divided by 3, which is 10. So sqrt(x) = 10*sqrt(6)

Now, to find x, we square both sides:

x = (10*sqrt(6))^2

Calculating that, 10 squared is 100, and (sqrt(6))^2 is 6. So 100*6 = 600. Therefore, x is 600.

Wait, let me verify each step again to make sure I didn't make a mistake. Starting with the inverse variation, y = k / sqrt(x). Plugging in x=24 and y=15: 15 = k / sqrt(24). Solving for k gives k = 15*sqrt(24). Then sqrt(24) is 2*sqrt(6), so k = 15*2*sqrt(6) = 30*sqrt(6). That seems right.

Then, using y=3, plug into y = 30*sqrt(6)/sqrt(x). So 3 = 30*sqrt(6)/sqrt(x). Multiply both sides by sqrt(x): 3*sqrt(x) = 30*sqrt(6). Divide by 3: sqrt(x) = 10*sqrt(6). Square both sides: x = (10*sqrt(6))² = 100*6 = 600. Yep, that checks out.

Alternatively, maybe there's a quicker way. Since y varies inversely with sqrt(x), then y1 * sqrt(x1) = y2 * sqrt(x2). Because in inverse variation, the product of the variables with their respective coefficients is constant. So, y1*sqrt(x1) = y2*sqrt(x2). Let's test that. 

Given y1=15, x1=24, y2=3, x2=?

So, 15*sqrt(24) = 3*sqrt(x2). 

Divide both sides by 3: 5*sqrt(24) = sqrt(x2)

Then square both sides: 25*24 = x2

25*24 is 600. So x2 is 600. Same answer. That's a quicker way. So that's another method confirming the result.

So both methods give x=600. Therefore, the answer should be 600.
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