Question
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Writing an equation that models variation w varies directly with \( u \) and inversely, with \( d \) In your equation, use \( k \) as the constant of proportionality. \( \square \) \( \square=\square \) \( \frac{\square}{\square} \)

Ask by Pritchard Ayala. in the United States
Mar 12,2025

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Answer

\( w = k \cdot \frac{u}{d} \)

Solution

To model the variation where \( w \) varies directly with \( u \) and inversely with \( d \), we can express this relationship mathematically. 1. **Direct Variation**: When we say \( w \) varies directly with \( u \), it means that as \( u \) increases, \( w \) also increases. This can be expressed as: \[ w = k \cdot u \] where \( k \) is the constant of proportionality. 2. **Inverse Variation**: When we say \( w \) varies inversely with \( d \), it means that as \( d \) increases, \( w \) decreases. This can be expressed as: \[ w = \frac{k}{d} \] Combining both direct and inverse variations, we can write the equation as: \[ w = k \cdot \frac{u}{d} \] Thus, the final equation that models the variation is: \[ w = k \cdot \frac{u}{d} \] In the format you requested: - \( w = k \cdot \frac{u}{d} \) - \( \square = k \cdot \frac{\square}{\square} \) Where: - The first square represents \( w \) - The second square represents \( u \) - The third square represents \( d \)

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The Deep Dive

To express the relationship where \( w \) varies directly with \( u \) and inversely with \( d \), you can write the equation as \( w = k \frac{u}{d} \). Here, \( k \) is the constant that links these variables together, reflecting the proportionality factors at play in this variation. When working with direct and inverse variations, one common mistake is forgetting to correctly place the variables in the formula. For direct variation, the variable should multiply, while for inverse variation, it should be in the denominator. Keep this in mind to avoid mixing the two types of relationships!

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