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
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
To find the relationship between M and N, we can set up a system of equations based on the given information. Let's call the coefficients of proportionality a and b:
1) M = aN + bN^2 (equation 1)
2) 1050 = a(10) + b(10)^2 (equation 2)
3) 1272 = a(12) + b(12)^2 (equation 3)
Simplifying equation 2 and 3, we have:
2) 1050 = 10a + 100b
3) 1272 = 12a + 144b
Now we have a system of two linear equations with two unknowns (a and b). We can solve this system to find the values of a and b.
Multiplying equation 2 by 12 and equation 3 by 10, we get:
4) 12600 = 120a + 1200b
5) 12720 = 120a + 1440b
Subtracting equation 4 from equation 5, we can eliminate a:
12720 - 12600 = 1440b - 1200b
120 = 240b
b = 120/240
b = 0.5
Now we substitute the value of b back into equation 2:
1050 = 10a + 100(0.5)
1050 = 10a + 50
1000 = 10a
a = 100/10
a = 10
Therefore, the relationship between M and N is:
M = 10N + 0.5N^2