Two quantities M and N are such that M varies partly as N and partly as the square of N.Determine the relationship between M and N given that when M=1050,N=10 and when M=1272,N=12

Answer 1

From the problem, we can write the relationship between M and N as:

M = kN + lN²

where k and l are some constants to be determined.

Using the first data point, when M = 1050 and N = 10, we can substitute these values into the equation above:

1050 = k(10) + l(10)²
1050 = 10k + 100l

Simplifying this equation, we get:

10k + 100l = 1050 ---> equation 1

Using the second data point, when M = 1272 and N = 12, we can substitute these values into the equation above:

1272= k(12) + l(12)²
1272 = 12k + 144l

Simplifying this equation, we get:

12k + 144l = 1272 ---> equation 2

Now, we have a system of two linear equations (equation 1 and equation 2) which we can solve simultaneously to find the values of k and l.

Multiplying equation 1 by 12 and equation 2 by 10, we get:

120k + 1200l = 12600 ---> equation 3
120k + 1440l = 12720 ---> equation 4

Subtracting equation 3 from equation 4, we obtain:

240l = 120 => l = 120 / 240 => l = 1/2

Now, substituting the value of l back into equation 1, we find:

10k + 100(1/2) = 1050
10k + 50 = 1050
10k = 1050 - 50
10k = 1000
k = 100

Thus, the relationship between M and N is:

M = 100N + (1/2)N²