Find a mathematical model representing the statement. (Determine the constant of proportionality.)
v varies jointly as p and q and inversely as the square of s. (v = 2.1 when p = 4.2, q = 5.4 and s = 2.4.)
v =
kpq/s^2 or v = kpq/s^2
To determine the constant of proportionality, substitute the given values into the equation:
2.1 = k(4.2)(5.4)/(2.4)^2
Simplifying the equation:
2.1 = k(22.68)/5.76
2.1 * 5.76 = 22.68k
12.096 = 22.68k
Divide both sides by 22.68:
k = 12.096 / 22.68
k ≈ 0.533
To find the mathematical model representing the statement, "v varies jointly as p and q and inversely as the square of s," we can use the formula for direct variation and inverse variation.
Direct Variation: When two quantities vary directly, one quantity is a constant multiple of the other. In this case, v varies directly with both p and q.
Inverse Variation: When two quantities vary inversely, their product is a constant. In this case, v varies inversely with the square of s.
Based on these relationships, we can write the mathematical model as:
v = k * (p * q) / (s^2)
Now, we need to find the constant of proportionality, k. To do this, we can use the given information: "v = 2.1 when p = 4.2, q = 5.4, and s = 2.4."
Substituting these values into the equation, we have:
2.1 = k * (4.2 * 5.4) / (2.4^2)
Simplifying this equation, we can calculate the value of k.
v = k*pq/s^2
Now just plug in the numbers to find k.