M is partly constant and partly varies as n square When N = 2 , M = 17 and when N= 4 , M= 53 . Find the connection and find n where M= 53

Answer 1

We can express the relationship between M and n as follows:

M = k * n^2 + c,

where k is the constant of variation and c is the constant.

From the given information, we have the following two equations when N = 2 and M = 17, and when N = 4 and M = 53.

17 = k * (2^2) + c,
53 = k * (4^2) + c.

Solving the system of equations, we get:

17 = 4k + c,
53 = 16k + c.

Subtract the first equation from the second:

53 - 17 = 16k - 4k + c - c,
36 = 12k,
k = 3.

Substitute the value of k back into the first equation:

17 = 4 * 3 + c,
17 = 12 + c,
c = 5.

So the relationship between M and n is given by:

M = 3n^2 + 5.

Now we can find the value of n when M = 53. Set M = 53 and solve for n:

53 = 3n^2 + 5,
48 = 3n^2,
16 = n^2.

Taking the square root of both sides:

n = ±4.

Since we already have the case where n = 4, the other solution is n = -4. So, n can be either 4 or -4 when M = 53.