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z varies directly as x squared. If z equals 48 when x equals 4, find z when x equals 5.

Question:

z varies directly as x squared. If z equals 48 when x equals 4, find z when x equals 5.

Direct Variation:

Direct variation describes two variables that are directly proportional or change in the same direction. When two variables are directly proportional, an increase in one variable causes an increase in the other variable and vice versa.

Answer and Explanation:

{eq}\displaystyle{ \\ }{/eq}

If {eq}z {/eq} varies directly as {eq}x^2 {/eq}, we can write this relationship as:

  • {eq}z\propto x^2 {/eq}

Adding a proportionality constant and removing the proportionality sign, we get:

  • {eq}z = kx^2 {/eq}

If {eq}z = 48 {/eq} when {eq}x = 4 {/eq}, then:

  • {eq}48 = (4)^2 k {/eq}
  • {eq}48 = 16k {/eq}

Solving for k:

  • {eq}k = \dfrac{48}{16} = 3 {/eq}

Therefore, the equation of variation is:

  • {eq}z = 3x^2 {/eq}

Using the above equation, the value of z' when {eq}x = 5 {/eq} is:

  • {eq}z = 3(5)^2 {/eq}
  • {eq}\boxed{\color{blue}{z = 75}} {/eq}

Learn more about this topic:

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Direct Variation: Definition, Formula & Examples

from ACT Prep: Help and Review

Chapter 13 / Lesson 7
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