Solving Proportionality Relations: Determining P When q and r Change

In mathematical relations, understanding the concept of proportionality is fundamental to solving a variety of problems. This article delves into the relationship between three variables, P, q, and r, where P varies directly as q and inversely as the square of r. We will solve a specific problem using the given values and demonstrate the process step-by-step.

Understanding Direct and Inverse Variation

When we say that P varies directly as q, it means that as q increases, P also increases proportionally. Mathematically, this can be represented as:

P ￡ q

On the other hand, when P varies inversely as the square of r, it implies that as r increases, P decreases, and the relationship is squared. This can be written as:

P ￡ 1/r2

Combining the Relations

By combining these two relations, we get the general form:

P ￡ q/r2

To express this in a more formal algebraic context:

P kq/r2

Here, k is the constant of proportionality. Our goal is to find this constant using the given values and then use it to solve for a new value of P when q and r change.

Given Data and Finding k

From the problem statement, we have the following values:

P 20

q 50

r 5

Substituting these into the relation:

20 k * 50 / 52

Simplify the equation:

20 k * 50 / 25

Solving for k:

20 k * 2

10 k

So, the constant of proportionality k is 10.

Using k to Find New Values of P

Now, we need to find the value of P when q 18 and r 6 . Using the previously found k 10 , we can substitute into our relation:

P 10 * 18 / 62

Simplifying:

P 10 * 18 / 36

P 10 * 1/2

P 5

Conclusion

This problem showcases the application of direct and inverse proportionality relations, a fundamental concept in algebra. By understanding the relationships between variables and utilizing the constant of proportionality, we can solve for unknown values in similar problems. This method is widely applicable in various fields, including physics, engineering, and economics.

Key Points

Direct variation: P ￡ q

Inverse variation: P ￡ 1/r2

General form: P kq/r2

Related Keywords

proportionality

direct and inverse variation

algebraic relations