C Variations Directly As P And Inversely As A: A Comprehensive Guide

Introduction

Mathematics is a subject that is often feared by many. However, when you understand the principles behind it, it can be quite simple. In this article, we will discuss the concept of c varying directly as p and inversely as a. This concept is often used in many fields, including physics, finance, and engineering. Understanding this concept will enable you to solve complex problems with ease.

What Does it Mean for C to Vary Directly as P?

When we say that c varies directly as p, we mean that as p increases, c increases as well. In other words, there is a direct proportionality between c and p. This can be represented mathematically as:

c = kp

Where k is a constant of proportionality.

What Does it Mean for C to Vary Inversely as A?

On the other hand, when we say that c varies inversely as a, we mean that as a increases, c decreases. In other words, there is an inverse proportionality between c and a. This can be represented mathematically as:

c = k/a

Where k is a constant of proportionality.

The Relationship Between Direct and Inverse Proportions

So if c varies directly as p and inversely as a, how do we put these two relationships together? The answer is simple: we multiply the direct relationship by the inverse relationship. This can be represented mathematically as:

c = kp/a

Example Problems

Let’s take a look at some example problems to help us better understand this concept:

Example 1:

If c varies directly as p and inversely as a, and c = 10 when p = 5 and a = 2, what is the value of c when p = 8 and a = 4?

Solution:

First, we need to find the value of k. We can do this by using the initial values of c, p, and a:

c = kp/a

10 = k(5)/2

k = 4

Now we can use k to find the value of c when p = 8 and a = 4:

c = kp/a

c = 4(8)/4

c = 8

Therefore, when p = 8 and a = 4, c = 8.

Example 2:

If c varies directly as p and inversely as the square root of a, and c = 20 when p = 10 and a = 4, what is the value of c when p = 5 and a = 16?

Solution:

First, we need to find the value of k. We can do this by using the initial values of c, p, and a:

c = kp/√a

20 = k(10)/√4

k = 10

Now we can use k to find the value of c when p = 5 and a = 16:

c = kp/√a

c = 10(5)/√16

c = 12.5

Therefore, when p = 5 and a = 16, c = 12.5.

Conclusion

Understanding the concept of c varying directly as p and inversely as a can be extremely useful in solving complex problems in various fields. By following the steps outlined in this article and practicing with example problems, you can master this concept and become a more confident problem solver.