$z$ varies directly as the square of $x$ and inversely as the cube of $y$. If $z=171$ when $x=8$ and $y=9$, find $z$ if $x=9$ and $y=6$. Round your answer to two decimal places if necessary.
Answer (2)
Write the general equation: z = k y 3 x 2 .
Solve for k using the given values z = 171 , x = 8 , and y = 9 : k = 171 ⋅ 8 2 9 3 = 64 124659 .
Substitute x = 9 , y = 6 , and k to find z : z = 64 124659 ⋅ 6 3 9 2 = 512 373977 .
Round the result to two decimal places: 730.42 .
Explanation
Write the equation We are given that z varies directly as the square of x and inversely as the cube of y . This can be written as: z = k y 3 x 2 where k is the constant of proportionality.
Substitute given values to find k We are given that z = 171 when x = 8 and y = 9 . We can use this information to find the value of k . Substituting these values into the equation, we get: 171 = k 9 3 8 2 171 = k 729 64
Solve for k Solving for k , we have: k = 171 ⋅ 64 729 k = 64 124659 k = 1947.796875
Substitute x=9 and y=6 to find z Now we want to find z when x = 9 and y = 6 . Substituting these values and the value of k into the equation, we get: z = 64 124659 ⋅ 6 3 9 2 z = 64 124659 ⋅ 216 81 z = 64 124659 ⋅ 8 3 z = 512 373977 z = 730.423828125
Round the answer Rounding to two decimal places, we get: z ≈ 730.42
Examples
Understanding direct and inverse variations is crucial in many real-world scenarios. For instance, in physics, the gravitational force between two objects varies directly with the product of their masses and inversely with the square of the distance between them. Similarly, in economics, the demand for a product might vary directly with consumer income and inversely with the price of the product. These relationships help us predict and understand how changes in one variable affect others.
To solve for z , we start with the variation equation z = k y 3 x 2 , find the constant k using the provided values, and then substitute new values to find z . After calculations, z is approximately 730.42 when x = 9 and $y = 6.
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