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Chapter 7: Problem 315

If \(p\) varies directly as \(q\) and \(p=9.6,\) when \(q=3\). find the equation that relates \(p\) and \(q\).

Short Answer

Expert verified
The equation that relates p and q is: \[ p = 3.2q \]

Step by step solution

01

Understand Direct Variation

Direct variation means that two variables, say p and q, are related in such a way that when one variable changes, the other variable changes proportionally. Mathematically, if p varies directly as q, this can be expressed as: \[ p = kq \]where k is the constant of variation.
02

Find the Constant of Variation

Given that p = 9.6 when q = 3, substitute these values into the direct variation equation to find the constant k. \[ 9.6 = k \times 3 \]Solving for k:\[ k = \frac{9.6}{3} \]\[ k = 3.2 \]
03

Write the Equation

Now that the constant of variation k is known, substitute k back into the direct variation equation. Thus, the equation that relates p and q is: \[ p = 3.2q \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

constant of variation
In mathematics, the constant of variation is a number that quantifies the ratio between two directly proportional variables. It is often represented by the letter \(k\). In direct variation, if two variables \(p\) and \(q\) are said to vary directly, their relationship can be expressed as \(p = kq\). This means that as \(q\) changes, \k\ times \(q\) gives the value of \(p\). For example, if \(p\) equals 9.6 when \(q\) is 3, we can find k by dividing 9.6 by 3. In this case, \[k = \frac{9.6}{3} = 3.2\]. Thus, k or the constant of variation is 3.2.
proportional relationships
Proportional relationships are those where two quantities change at the same rate. These relationships can be described by direct variation. In our example, \(p\) and \(q\) have a proportional relationship, which means that \(p = 3.2q\). Here鈥檚 why this is important: When \(q\) increases, \(p\) also increases by a factor of 3.2 and if \(q\) decreases, \(p\) decreases proportionally. This can be useful in many practical situations. For instance, if you know the proportionate relationship between time worked and payment, you can easily calculate the earnings for any amount of time worked.
linear equations
A linear equation is a type of equation where the highest exponent of the variable is one. In the case of direct variation, the equation \(p = kq\) is a linear equation because both \(p\) and \(q\) are to the power of one. This equation forms a straight line when graphed, showing a constant rate of change. Each point on the line represents a pair \(q, p\) that satisfies the equation. Understanding this helps in visualizing how changes in \(q\) affect \(p\) consistently. For instance, plotting the equation we derived (i.e., \(p = 3.2q\)) on a coordinate plane would result in a straight line that passes through the origin (0,0) and follows the consistent ratio of 3.2.

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