X varies directly as a product of u and v and inversely s thoer sum . If x=3 when u=3andv=1 find (a) the relationship connecting x,u,v (b). What is the value of x if u=3andv=3?
x = k u v/(u+v)
3 = k 3*1 /(4)
so
k = 4
so
x = 4 u v / (u+v)
so put in 3 for u and 3 for v
4*9/6
The answer is 21
To find the relationship connecting x, u, and v, we need to consider the given information. We are told that x varies directly as the product of u and v and inversely as their sum. Mathematically, we can represent this relationship as:
x ∝ (u * v) / (u + v)
Now, we can proceed to solve the problem.
(a) To find the relationship connecting x, u, and v, we can use the given values. We are told that when u = 3 and v = 1, x = 3. Substituting these values into our equation, we get:
3 ∝ (3 * 1) / (3 + 1)
3 ∝ 3/4
Now, to find the constant of proportionality, we can cross-multiply:
3 * 4 = 3
12 = 3
Hence, the relationship connecting x, u, and v is x = (3/4) * (u * v) / (u + v) or x = (3u * v) / (4u + 4v).
(b) Now, we need to find the value of x when u = 3 and v = 3. We can use the relationship we found in part (a) to calculate this value:
x = (3/4) * (3 * 3) / (3 + 3)
x = (3/4) * 9 / 6
x = (27/4) / 6
Simplifying further:
x = 27/24
Finally, x = 9/8.
Therefore, when u = 3 and v = 3, the value of x is 9/8.