To express the relationship where y varies directly as x and inversely as the square of z, we can use the concept of proportionality in mathematics.
When we say that y varies directly as x, it means that as x increases, y also increases. This can be expressed as:
y = k * x
where k is a constant of proportionality.
On the other hand, saying that y varies inversely as the square of z means that as z increases, y decreases in relation to the square of z. This can be expressed mathematically as:
y = k’ / (z²)
where k’ is another constant of proportionality.
To combine both relationships, we can express y in terms of both x and z:
y = k * (x / z²)
In this equation, y is directly proportional to x and inversely proportional to the square of z. The constant k captures the combined relationship. This expression highlights how changes in x and z will affect the value of y.
In summary, the relationship can be succinctly expressed as:
y = k * (x / z²)
This formula is valuable in various fields including physics, economics, and engineering, where understanding how variables interact is essential.