Respuesta :
[tex]\qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad \stackrel{\textit{constant of variation}}{y=\stackrel{\downarrow }{k}x~\hfill } \\\\ \textit{\underline{x} varies directly with }\underline{z^5}\qquad \qquad \stackrel{\textit{constant of variation}}{x=\stackrel{\downarrow }{k}z^5~\hfill } \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]
[tex]\stackrel{\textit{"y" varies directly as "x"}}{y = kx}\qquad \textit{we also know that} \begin{cases} y = 540\\ x = 10 \end{cases}\implies 540=k(10) \\\\\\ \cfrac{540}{10}=k\implies 54=k~\hfill \boxed{y=54x} \\\\\\ \textit{when y = 1080, what is "x"?}\qquad 1080=540x\implies \cfrac{1080}{540}=x\implies 2=x[/tex]
The value of x when y = 1080 for the condition when its given that y varies directly as x and that y = 540 when x = 10, is found to be x = 20
What is directly proportional and inversely proportional relationship?
Let there are two variables p and q
Then, p and q are said to be directly proportional to each other if
[tex]p = kq[/tex]
where k is some constant number called constant of proportionality.
This directly proportional relationship between p and q is written as
[tex]p \propto q[/tex] where that middle sign is the sign of proportionality.
In a directly proportional relationship, increasing one variable will increase another.
Now let m and n are two variables.
Then m and n are said to be inversely proportional to each other if
[tex]m = \dfrac{c}{n} \\\\ \text{or} \\\\ n = \dfrac{c}{m}[/tex]
(both are equal)
where c is a constant number called constant of proportionality.
This inversely proportional relationship is denoted by
[tex]m \propto \dfrac{1}{n} \\\\ \text{or} \\\\n \propto \dfrac{1}{m}[/tex]
As visible, increasing one variable will decrease the other variable if both are inversely proportional.
For this case, it is specified that y varies directly as x.
That means
[tex]y \propto x\\or\\y = kx[/tex]
At x = 10, y is 540. Putting these values in the equation y = kx as they're related, we get:
[tex]540 = k \times 10\\\\\text{Dividing both the sides by 10}\\\\\dfrac{540}{10} = k\\\\k = 54[/tex]
Thus, the relationship between x and y is: y = 54x
Now, when y = 1080, we get:
[tex]1080 = 54x\\\\\text{Dividing both the sides by 54}\\\\\dfrac{1080}{54} = x\\\\x = 20[/tex]
Thus, the value of x when y = 1080 for this condition is when x = 20.
Learn more about directly related variables here:
https://brainly.com/question/13082482
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