25. [tex]$z$[/tex] varies inversely with [tex]$x$[/tex] and directly with [tex]$y$[/tex]. When [tex]$x=6$[/tex] and [tex]$y=2, z=5$[/tex].
What is the value of [tex]$z$[/tex] when [tex]$x=4$[/tex] and [tex]$y=9$[/tex]?
Answer (2)
Write the equation relating z , x , and y using a constant of proportionality k : z = k x y .
Substitute the given values x = 6 , y = 2 , and z = 5 into the equation to solve for k : 5 = k 6 2 .
Solve for k : k = 5 ⋅ 2 6 = 15 .
Substitute the new values x = 4 and y = 9 into the equation to find the value of z : z = 15 4 9 = 4 135 .
Explanation
Setting up the equation We are given that z varies inversely with x and directly with y . This means that z is proportional to x y . We can write this relationship as: z = k x y where k is the constant of proportionality.
Finding the constant of proportionality We are given that when x = 6 and y = 2 , z = 5 . We can use this information to find the value of k . Substituting these values into the equation, we get: 5 = k 6 2
Calculating k To solve for k , we multiply both sides of the equation by 2 6 : k = 5 ⋅ 2 6 = 5 ⋅ 3 = 15 So, k = 15 .
Substituting new values Now we have the equation: z = 15 x y We want to find the value of z when x = 4 and y = 9 . Substituting these values into the equation, we get: z = 15 4 9
Calculating z Calculating the value of z , we have: z = 4 15 × 9 = 4 135 = 33.75
Examples
Understanding direct and inverse variations is crucial in many real-world scenarios. For example, in physics, the speed of an object is directly proportional to the distance it travels and inversely proportional to the time it takes. If you know the relationship between these variables, you can predict how changing one variable will affect the others. This type of problem-solving is also applicable in economics, where demand might be inversely proportional to price, or in engineering, where the strength of a material might be directly proportional to its thickness.
The value of z when x = 4 and y = 9 is 33.75. This was determined using the equation z = k x y , where k was found to be 15 from initial conditions. By substituting the new values into the equation, the result was calculated as z = 33.75 .
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