Find the constant of variation [tex]$k$[/tex] for the inverse variation. Then write an equation for the inverse variation.
[tex]$y=5 \text { when } x=7.5$[/tex]
Answer (2)
We recognize that y varies inversely with x , so y = x k .
We substitute the given values y = 5 and x = 7.5 into the equation to find k : 5 = 7.5 k .
We solve for k : k = 5 × 7.5 = 37.5 .
We write the equation for the inverse variation: y = x 37.5 . The final answer is y = x 37.5 .
Explanation
Understanding Inverse Variation We are given that y = 5 when x = 7.5 and that y varies inversely with x . This means that as x increases, y decreases, and vice versa. The relationship between x and y can be expressed as y = x k , where k is the constant of variation. Our goal is to find the value of k and then write the equation for the inverse variation.
Finding the Constant of Variation To find the constant of variation k , we substitute the given values of x and y into the equation y = x k . We have y = 5 and x = 7.5 , so we can write: 5 = 7.5 k To solve for k , we multiply both sides of the equation by 7.5 :
k = 5 × 7.5 k = 37.5
Writing the Inverse Variation Equation Now that we have found the constant of variation k , we can write the equation for the inverse variation. The equation is given by y = x k , and we know that k = 37.5 , so the equation is: y = x 37.5
Final Answer Therefore, the constant of variation is k = 37.5 , and the equation for the inverse variation is y = x 37.5 .
Examples
Inverse variation is a concept that appears in many real-world situations. For example, the time it takes to complete a journey is inversely proportional to the speed at which you travel. If you double your speed, you halve the time it takes to reach your destination. Similarly, in physics, the pressure of a gas is inversely proportional to its volume (at constant temperature). Understanding inverse variation helps us make predictions and understand relationships in various fields, from travel planning to scientific research.
The constant of variation k is found to be 37.5, and the equation for the inverse variation is y = x 37.5 .
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