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The variable z varies jointly with y and the square of x. If x= -2 when y= 7 and z= -84, find x...

Question:

The variable z varies jointly with y and the square of x. If x= -2 when y= 7 and z= -84, find x when z= -96 and y= 2.

Joint Variation:

If a variable {eq}c {/eq} varies jointly with respect to both the variables {eq}a {/eq} and {eq}b {/eq}, then it means that:

$$c= kab $$

where, {eq}k {/eq} is a constant of variation.

Note: It means that {eq}c {/eq} is directly proportional to each of {eq}a {/eq} and {eq}b {/eq} taken one at a time.

Answer and Explanation:

The variable {eq}z {/eq} varies jointly with {eq}y {/eq} and square of {eq}x {/eq}.

So we get:

$$z= k y x^2 \,\,\,\,\,\,\rightarrow (1) $$

Substitute the first set of given values {eq}x= -2, y= 7 \text{ and }z= -84 {/eq} in (1):

$$-84 = k (7) (-2)^2 \\ -84 = k (7)(4) \\ -84 = 28k \\ \text{Dividing both sides by 28}, \\ k=-3 $$

Substitute this and another set of given values {eq}z=-96 {/eq} and {eq}y=2 {/eq} in (1):

$$-96 = -3 (2)(x^2) \\ -96 = -6x^2 \\ \text{Dividing both sides by -6}, \\ x^2=16 \\ \text{Taking square root on both sides}, \\ x= \boxed{\mathbf{\pm 4}} $$


Learn more about this topic:

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Constant of Variation: Definition & Example

from High School Algebra II: Tutoring Solution

Chapter 19 / Lesson 10
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