Equations, Inequalities and VICs (3rd ed), Chapter 5, p. 95

[nvingers]

In Action Problem 18:

The solution to question 18 (bottom of p.95) states that "[t]he formula shows that Q is directly proportional to w and inversely proportaional to x and to z^2."

How does the formula Q = 5w / 4xz^2 show that Q is directly proportional to w and inversely proportaional to x and to z^2?

Please help!

[esledge]

"Directly proportional" means that when one variable increases, the other does too, in the same ratio. Saying that "Q is directly proportional to w" means that doubling w has the effect of doubling Q. Mathematically, this is Q = kw, where k is a constant.

"Inversely proportional" means that when one variable increases, the other decreases, in the inverse ratio. Saying that "Q is inversely proportional to x" means that doubling x has the effect of halving Q. Mathematically, this is Q = k/x, where k is a constant.

The constant k is relatively unimportant. What really matters is the relationship between the variables. Here's how to remember it:

directly proportional --> both variables in the numerator, on either side of the equal sign.
inversely proportional --> one variable in the numerator on the left side of the equation, the other in the denominator on the right side (or vice-versa).

How does the formula Q = 5w / 4xz^2 show that Q is directly proportional to w and inversely proportaional to x and to z^2?

Since Q and w are across from each other in the numerators, Q and w are directly proportional.
Since Q is in the numerator while x and z^2 are in the denominator on the other side of the equation, Q is inversely proportional to both x and z^2.

[mozart611]

I'm still not clear, especially about how tripling z causes z^2 to increase by a factor of 9. If z = 2, tripling z will make it 2^3 = 8, 8^2 = 64, which is not a factor of 9. Am I missing something?

[Ben Ku]

When we triple a number, we MULTIPLY it by 3, not raise it to the third power. So if z = 2, then 3z = 6. So while z^2 = 4, (3z)^2 = (6)^2 = 36 = 4*9.

Hope that helps.