x is partly constant and partly varies as y^2 when y =11,x=52.8and when y = 5 and x = 8 find (a) equation connecting x and y (b) y when x = 105.6
x = ky^2 + m
52.8 = 121k+m
8 = 25k+m
43.2 = 96k
Now get k, and use that to find m.
Then use k and m to find y when x=105.6
To find the equation connecting x and y in this scenario, we need to first determine the two parts of x: the constant part and the part that varies with y^2.
Given that x is partly constant and partly varies as y^2, we can express this as:
x = c + ky^2
where c is the constant part and k is the coefficient that represents how x varies with y^2.
Now, let's use the given information to find the values of c and k.
When y = 11 and x = 52.8, we can substitute these values into the equation:
52.8 = c + k(11)^2
52.8 = c + 121k
Similarly, when y = 5 and x = 8, we can substitute these values into the equation:
8 = c + k(5)^2
8 = c + 25k
We now have a system of two equations:
52.8 = c + 121k ---(1)
8 = c + 25k ---(2)
To solve this system, we can subtract equation (2) from equation (1) to eliminate c:
52.8 - 8 = c + 121k - c - 25k
44.8 = 96k
Simplifying the equation, we find:
k = 44.8 / 96
k ≈ 0.4667
Now let's substitute the value of k back into equation (2) to find c:
8 = c + 25(0.4667)
8 = c + 11.6675
Subtracting 11.6675 from both sides, we get:
c = 8 - 11.6675
c ≈ -3.6675
So, the equation connecting x and y is:
x = -3.6675 + 0.4667y^2
That answers part (a) of the question.
Now, to find y when x = 105.6, we can rearrange the equation as follows:
105.6 = -3.6675 + 0.4667y^2
Adding 3.6675 to both sides:
105.6 + 3.6675 = 0.4667y^2
109.2675 = 0.4667y^2
Dividing both sides by 0.4667:
y^2 ≈ 233.9091
Taking the square root of both sides:
y ≈ √233.9091
y ≈ 15.285
Therefore, when x = 105.6, y is approximately equal to 15.285. This answers part (b) of the question.