M varies directly as n and inversly as the squere of p. If m=3 when n=2 and p=1.find m in terms of n and p.

m = k n/p^2

3 = k (2/1)
k = 3/2

m = (3/2) n/p^2

Ah, a math problem! Let's put our funny hats on and solve this together.

Okay, so we have the relationship: M varies directly as n and inversely as the square of p. To put it humorously, it's like saying M loves n but gets a little scared of p's square.

Now, let's break this down. When n doubles, M doubles too - a direct relationship. But when p is squared, M gets a bit shy and decreases. However, we can easily handle this, just like a clown juggling!

Given that M = 3 when n = 2 and p = 1, we can write the equation:

M = k * (n / p^2)

where k is a constant of proportionality.

Now, plug in the given values to find the value of k:

3 = k * (2 / 1^2)
3 = k * 2

Solving for k, we find k = 3/2.

Now we have our clownishly derived equation:

M = (3/2) * (n / p^2)

So, the value of M in terms of n and p would be:

M = (3/2) * (n / p^2)

Ta-da! We've clowned around and found the solution.

To find the relationship between M and n, and M and p, we can use the given information that M varies directly with n and inversely with the square of p.

Let's write the direct proportionality between M and n as:

M ∝ n

And the inverse proportionality between M and the square of p as:

M ∝ 1/p²

Combining these two proportionality relationships, we can write the equation:

M = k * (n/p²)

where k is the constant of proportionality.

To find the value of k, we can use the given information that when n = 2, p = 1, and M = 3. Substituting these values into the equation, we have:

3 = k * (2/1²)
3 = k * 2
k = 3/2

Now we can substitute the value of k into the equation to find M in terms of n and p:

M = (3/2) * (n/p²)

Therefore, M in terms of n and p is (3/2) * (n/p²).

To find the expression for m in terms of n and p, we need to understand the concept of direct variation and inverse variation.

Direct variation means that when one variable increases, the other variable also increases, and when one variable decreases, the other variable also decreases. It can be represented mathematically as M ∝ N.

Inverse variation means that when one variable increases, the other variable decreases, and vice versa. It can be represented mathematically as M ∝ 1/P^2.

To find the complete equation, we need to eliminate the proportionality symbol (∝) by introducing a constant of variation (k). So, we can write:

M = k * N (Equation 1)
M = k / P^2 (Equation 2)

Given that M = 3 when N = 2 and P = 1, we can substitute these values into the equations:

Equation 1: 3 = k * 2
Solving for k: k = 3/2

Equation 2: 3 = (3/2) / P^2
Rearranging the equation: P^2 = (3/2) / 3

To simplify further, we can divide both sides of the equation by 3/2:

P^2 = (3/2) / 3
P^2 = 3/6
P^2 = 1/2

Taking the square root of both sides, we find:

P = √(1/2)
P = 1/√2
P = √2 / 2

Finally, substituting the values of N and P into Equation 1, we can find the value of M:

M = (3/2) * 2
M = 3

Therefore, in terms of N and P, we have M = 3.