The variable $y$ is inversely proportional to variable $x$. If $y=$ 0.4 when $x=2.5$, find the value of the constant of proportionality.
Answer (2)
Recognize that inverse proportionality means y = x k .
Substitute the given values x = 2.5 and y = 0.4 into the equation.
Solve for k by multiplying both sides by 2.5 .
Find that the constant of proportionality is 1 .
Explanation
Understanding Inverse Proportionality We are given that y is inversely proportional to x . This means that there exists a constant k such that y = x k . We are also given that when x = 2.5 , y = 0.4 . Our goal is to find the value of k .
Substituting Values To find the constant of proportionality k , we can substitute the given values of x and y into the equation y = x k . So we have: 0.4 = 2.5 k
Solving for k Now, we solve for k by multiplying both sides of the equation by 2.5 :
k = 0.4 × 2.5
Calculating k Calculating the value of k , we get: k = 1.0
Final Answer Therefore, the constant of proportionality is 1 .
Examples
Inverse proportionality appears in various real-world scenarios. For instance, the time it takes to complete a journey is inversely proportional to the speed. If you double your speed, you halve the time it takes to reach your destination, assuming the distance remains constant. Similarly, in physics, the current flowing through a resistor is inversely proportional to the resistance when the voltage is kept constant. Understanding inverse proportionality helps in making informed decisions in everyday life, such as planning travel times or understanding electrical circuits.
The constant of proportionality k for the given values is 1 . This is derived from the inverse proportionality relationship y = x k , substituting the values provided leads to the conclusion that k = 1.0 . Understanding this principle helps in various practical applications in mathematics and real life.
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