A varies directly as x and B is inversely proportional to x^2. Given that y= A+B and that y= 19 when x=2 or 3, express y in terms of x.
A = mx
B = n/x^2
y = A+B = mx + n/x^2
19 = 2m + n/4
19 = 3m + n/9
m=5
n=36
y = 5x + 36/x^2
Thanks!
To express y in terms of x, we need to find the relationships between A and B with respect to x using the given information.
Let's start by analyzing the relationship between A and x. We are given that A varies directly with x, which means that A is directly proportional to x. In mathematical terms, we can write this as:
A = kx,
where k is a constant of proportionality that we need to determine.
Next, let's consider the relationship between B and x^2. We are told that B is inversely proportional to x^2, which means that B is inversely proportional to the square of x. In mathematical terms, we can represent this as:
B = k' / x^2,
where k' is another constant of proportionality that we need to find.
Now, let's substitute the expressions for A and B into the equation for y:
y = A + B
Since we now have expressions for A and B, we can substitute those in:
y = kx + k' / x^2
To determine the values of k and k', we can use the given information that y equals 19 when x is 2 or 3:
When x = 2:
19 = k(2) + k' / (2^2) -> 19 = 2k + k' / 4
When x = 3:
19 = k(3) + k' / (3^2) -> 19 = 3k + k' / 9
Now, we have a system of two equations with two variables (k and k'), which we can solve to find their values.
To solve this system of equations, we can use either substitution or elimination method:
Let's eliminate k' by multiplying the first equation by 9 and the second equation by 4:
36k = 9(19) - k'
4(19) = 3k + k' / 9
Now, we can add the two equations to eliminate k':
36k + 4(19) = 171
36k = 171 - 76
36k = 95
k = 95/36
Substituting the value of k back into the first equation, we can solve for k':
19 = (95/36)(2) + k' / 4
19 = 190/36 + k' / 4
19 = 95/18 + k' / 4
(84/18)(4) = k'
k' = 112/3
Now that we have the values of k and k', let's substitute them back into the equation for y:
y = (95/36)x + (112/3) / x^2
Therefore, y in terms of x is y = (95/36)x + (112/3) / x^2.