X varies directly as the product of u and v and inversely as their sum if x=3 when u=3 and x=1 what is the value of x if u=3 and v=3?
X = u*v/(u+v)
X = 3*3/(3+3) = 9/6 = 1.5.
X varies directly as the product of u and v and inverselyas their sum. If x =3 when u =3 and v =1 what is thevalue of x if u =3 when u = 3 and v = 3?
To solve this problem, we need to determine the relationship between x, u, and v.
The given information states that x varies directly as the product of u and v and inversely as their sum. This can be written as:
x ∝ (u * v) / (u + v)
We are given that x = 3 when u = 3. We can substitute these values into the equation to find the constant of proportionality (k):
3 = k * (3 * v) / (3 + v)
Simplifying, we have:
3 * (3 + v) = k * 3v
9 + 3v = 3kv
9 = 3kv - 3v
9 = v(3k - 3)
v = 9 / (3k - 3)
v = 3 / (k - 1)
Now, we can use the second given condition: x = 1 when u = 3 and v = 3. We can substitute these values into the equation and solve for k:
1 = k * (3 * 3) / (3 + 3)
1 = k * 9 / 6
6 = 9k
k = 6 / 9
k = 2/3
Now that we have the value of k, we can substitute the values of u = 3 and v = 3 into the equation to find x:
x = (u * v) / (u + v)
x = (3 * 3) / (3 + 3)
x = 9 / 6
x = 3/2 or 1.5
Therefore, the value of x when u = 3 and v = 3 is 1.5.
To solve this problem, we can use the direct and inverse variation formulas.
The direct variation formula states that if two variables, X and Y, vary directly, then X = kY. In this case, X varies directly as the product of u and v, so we can write it as X = k(uv).
The inverse variation formula states that if two variables, X and Y, vary inversely, then XY = k. In this case, X varies inversely as the sum of u and v, so we can write it as X(u + v) = k.
To find the value of X when u = 3 and v = 3, we can substitute these values into the direct and inverse variation formulas and solve for X.
First, let's use the given information to find the value of k:
When x = 3 and u = 3:
3 = k(3 * v)
3 = 3kv
k = 1/v
Now, let's use the inverse variation formula to find the value of X:
X(u + v) = k
X(3 + 3) = 1/v
X(6) = 1/v
X = 1/(6v)
Substituting v = 3:
X = 1/(6 * 3)
X = 1/18
Therefore, when u = 3 and v = 3, the value of X is 1/18.