5. If m varies directly as n and inversely as the square of p, and m = 2 when when n=18 and p=2,find m when p=3 and n=25
"m varies directly as n and inversely as the square of p"
---> m = k(n/p^2)
m = 2 when when n=18 and p=2
---> 2 = k(18/4)
k = 8/18 = 4/9
m = (4/9)(n/p^2)
when p=3 and n=25
m = (4/9)(25/9) = 100/81
To find the value of m when p = 3 and n = 25, we can use the direct variation and inverse variation equations.
The direct variation equation states that m varies directly as n, so we can write it as:
m ∝ n
The inverse variation equation states that m varies inversely as the square of p, so we can write it as:
m ∝ 1 / p^2
Combining these two equations, we have:
m ∝ n / p^2
To find the constant of variation, we can substitute the given values of m, n, and p:
m = 2, n = 18, and p = 2
2 = (18) / (2^2)
2 = 18 / 4
Now we can solve for the constant of variation:
2 = 4.5
Therefore, the constant of variation is 4.5.
Now we can use this constant to find m when p = 3 and n = 25:
m = (25) / (3^2)
m = 25 / 9
m ≈ 2.778
Therefore, when p = 3 and n = 25, m is approximately equal to 2.778.
To solve this problem, we can use the concept of direct and inverse variation.
1. We are given that m varies directly as n and inversely as the square of p. This can be written as:
m = k * (n / p^2)
Where k is the constant of variation.
2. We are also given a specific scenario where m = 2 when n = 18 and p = 2. Let's substitute these values into the equation and solve for k:
2 = k * (18 / 2^2)
2 = k * (18 / 4)
2 = k * 4.5
k = 2 / 4.5
k ≈ 0.4444
So, the constant of variation is approximately 0.4444.
3. Now that we know the value of k, we can use it to find m when p = 3 and n = 25. Substitute these values into the equation:
m = 0.4444 * (25 / 3^2)
m = 0.4444 * (25 / 9)
m = 0.4444 * 2.7778
m ≈ 1.2346
Therefore, when p = 3 and n = 25, the value of m is approximately 1.2346.