If there is inverse variation and [tex]y=-4[/tex] when [tex]x=14[/tex], find [tex]x[/tex] when [tex]y=28[/tex].
A. 2
B. -2
C. 98
D. -98
Answer (1)
The problem states that x and y vary inversely, so x y = k for some constant k .
We find k using the given values x = 14 and y = − 4 : k = 14 × − 4 = − 56 .
We substitute y = 28 into the equation x y = − 56 to get x × 28 = − 56 .
Solving for x , we find x = 28 − 56 = − 2 , so the final answer is − 2 .
Explanation
Finding the Constant of Variation We are given that x and y vary inversely. This means that their product is a constant. We can write this relationship as x y = k , where k is the constant of variation. We are given that when x = 14 , y = − 4 . We can use this information to find the value of k .
Calculating k Substitute x = 14 and y = − 4 into the equation x y = k to find k :
k = ( 14 ) × ( − 4 ) = − 56
Substituting y = 28 Now we know that the relationship between x and y is x y = − 56 . We want to find the value of x when y = 28 . Substitute y = 28 into the equation and solve for x :
x × ( 28 ) = − 56
Solving for x To solve for x , divide both sides of the equation by 28:
x = 28 − 56 = − 2
Final Answer Therefore, when y = 28 , x = − 2 .
Examples
Inverse variation is a concept that appears in many real-world scenarios. 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, assuming the distance remains constant. Similarly, in physics, the pressure of a gas is inversely proportional to its volume at a constant temperature. Understanding inverse variation helps in making predictions and optimizing processes in various fields.
