y is partly constant and partly varies as the cube of x given that whenx=1,y=5,andx=2,y=-1 find the relationship betweenx and y hence find the value of y when x=3
To find the relationship between x and y, we can start by assuming that y can be represented as a linear function of x and the cube of x. Let's say:
y = ax + bx^3
We are given two data points: when x = 1, y = 5, and when x = 2, y = -1. We can use these data points to solve for the values of a and b.
1. For the first data point, x = 1 and y = 5. Let's substitute these values into the equation:
5 = a(1) + b(1)^3
5 = a + b
2. For the second data point, x = 2 and y = -1. Substitute these values into the equation:
-1 = a(2) + b(2)^3
-1 = 2a + 8b
Now we have two equations:
Equation 1: 5 = a + b
Equation 2: -1 = 2a + 8b
Solving these equations simultaneously will give us the values of a and b.
Multiply Equation 1 by 2:
2(5) = 2a + 2b
10 = 2a + 2b
Subtract Equation 2 from the above equation:
10 - (-1) = 2a + 2b - (2a + 8b)
11 = -6b
Divide both sides by -6:
b = -11/6
Substitute the value of b back into Equation 1:
5 = a + (-11/6)
30/6 + 11/6 = a
41/6 = a
So, the relationship between x and y is:
y = (41/6)x - (11/6)x^3
Now, to find the value of y when x = 3, substitute x = 3 into the equation:
y = (41/6)(3) - (11/6)(3)^3
Simplifying this equation will give us the value of y when x = 3.