If [tex]$x$[/tex] varies inversely as [tex]$y$[/tex] and [tex]$x=10$[/tex] when [tex]$y=15$[/tex], find the value of [tex]$y$[/tex] when [tex]$x=25$[/tex].
A. 6
B. [tex]$16^2 / 3$[/tex]
C. 25
D. [tex]$371 / 2$[/tex]
Answer (2)
When x = 25 , the value of y is 6 after establishing the constant of inverse proportionality as 150 using the initial conditions.
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Establish the inverse variation relationship: x = y k .
Use given values x = 10 and y = 15 to find the constant k : k = 10 × 15 = 150 .
Substitute x = 25 into the equation x = y 150 .
Solve for y : y = 25 150 = 6 . The final answer is 6 .
Explanation
Understanding the Problem We are given that x varies inversely as y , which means that x and y are related by the equation x = y k , where k is a constant of proportionality. We are also given that x = 10 when y = 15 . Our goal is to find the value of y when x = 25 .
Finding the Constant of Proportionality First, we need to find the constant of proportionality k . We can use the given values x = 10 and y = 15 to find k . Substituting these values into the equation x = y k , we get: 10 = 15 k
Calculating the Constant To solve for k , we multiply both sides of the equation by 15: k = 10 × 15 = 150 So, the constant of proportionality is k = 150 .
Setting up the Equation Now we have the equation relating x and y : x = y 150 We want to find the value of y when x = 25 . Substitute x = 25 into the equation: 25 = y 150
Solving for y To solve for y , we can multiply both sides of the equation by y and then divide by 25: 25 y = 150 y = 25 150 = 6 Therefore, when x = 25 , the value of y is 6.
Final Answer The value of y when x = 25 is 6.
Examples
Imagine you're adjusting the speed of a fan. If you decrease the fan's speed (x), the time it takes to cool the room (y) increases. This inverse relationship helps you understand how changing one variable affects the other. In real life, this concept is useful in various scenarios, such as adjusting the aperture of a camera to control the amount of light or managing the speed of a vehicle to optimize fuel consumption. Understanding inverse variation allows for efficient adjustments and predictions in everyday situations.
