What is the constant of variation, $k$, of the direct variation, $y=k x$, through $(5,8)$?
The problem describes a direct variation relationship defined by the equation \(y = kx\), where \(k\) is the constant of variation. We are given that this relationship passes through the point \((x, y) = (5, 8)\). Our goal is to determine the value of \(k\).
To find the constant \(k\), we substitute the given values \(x = 5\) and \(y = 8\) into the direct variation equation \(y = kx\). This gives us:
\[8 = k \times 5\]
To isolate \(k\), we divide both sides of the equation by \(5\):
\[k = \frac{8}{5}\]
Thus, the constant of variation is \(\frac{8}{5}\). Comparing this result to the provided options, we see that it matches the last choice.
Direct variation is a fundamental concept for understanding proportional relationships. For example, if you are shopping for apples and notice that the total cost \(y\) is directly proportional to the number of apples \(x\) you purchase, you are dealing with a direct variation. If 5 apples cost \(\$8\), the constant of variation \(k = \frac{8}{5} = 1.6\) represents the price per apple in dollars. Knowing this constant helps you quickly calculate the total cost for any number of apples.