Pregunta
6. If \( Y \) varies inversely as \( X^{2} \) and \( X \) varies directly as \( Z^{2} \). Find the relationship between \( Y \) and \( Z \) if \( C \) is a constant.
Ask by Greene Daniels. in Nigeria
Feb 18,2025
Solución de inteligencia artificial de Upstudy
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Y varies inversely as Z⁴.
Solución
We are given two relationships:
1. Y varies inversely as X², which means:
Y = K / X² (Equation 1)
where K is a constant.
2. X varies directly as Z², which gives:
X = L · Z² (Equation 2)
where L is another constant.
Substitute Equation 2 into Equation 1:
Y = K / (L · Z²)²
Y = K / (L² · Z⁴)
We can combine the constants K and L² into one constant, say C:
Y = C / Z⁴ (Where C = K / L²)
Thus, the relationship between Y and Z is that Y varies inversely as Z⁴.
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Simplifique esta solución The Deep Dive
Since \( Y \) varies inversely as \( X^{2} \), we can express this relationship as \( Y = \frac{C_1}{X^{2}} \), where \( C_1 \) is a constant. Given that \( X \) varies directly as \( Z^{2} \), we can write that relationship as \( X = C_2 Z^{2} \), where \( C_2 \) is another constant. By substituting \( X \) from the second equation into the first equation, we get: \[ Y = \frac{C_1}{(C_2 Z^{2})^{2}} = \frac{C_1}{C_2^2 Z^{4}} \] Thus, we can conclude that \( Y \) varies inversely as \( Z^{4} \).
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