1. the current i varies directly as the electromotive force E and inversely as the resistance R. If in a system a current of 20 amperes flows through a resistance of 20 ohms with an electromotive force of 100 volts, find the current that 150 volts will send through the system.

Variation is a relationship of two or more variables. We say that the two variables vary directly when these variables go in the same direction. Meaning, if one variable increases, the other one also increases. We say that the two variables vary inversely when these variables do not go in the same direction.

Types of Variation

1. Varies Directly

2. Varies Inversely

3. Joint Variation

4. Combine Variation

Suppose a variable, [tex]y[/tex], varies directly as [tex]x[/tex], then there exist a constant, [tex]k[/tex], such that [tex]y=kx[/tex] satisfies.

Suppose a variable, [tex]y[/tex], varies inversely as [tex]x[/tex], then there exist a constant, [tex]k[/tex], such that [tex]y=\frac{k}{x}[/tex] satisfies.

In the context of the problem, since the current, [tex]i[/tex], varies directly as the electromotive force [tex]E[/tex], then we know that there is a constant, [tex]k[/tex], such that [tex]i=kE[/tex] satisfies. Also, since current, [tex]i[/tex], varies inversely as the resistance [tex]R[/tex]. Simultaneously, we have [tex]i=\frac{kE}{R}[/tex].

First, solve the value of [tex]k[/tex] by substituting [tex]i=20[/tex], [tex]R=20[/tex], and [tex]E=100[/tex] into the formula.

[tex]\begin{aligned}i&=\frac{kE}{R}\\20&=\frac{k(100)}{20}\\20&=k(5)\\\frac{20}{5}&=\frac{k(5)}{5}\\4&=k\end{alinged}[/tex]

Second, using the same formula, substitute the value of [tex]k=4[/tex], [tex]E=150[/tex], and [tex]R=20[/tex] to find the current.

[tex]\begin{aligned}i&=\frac{4(150)}{20}\\i&=\frac{600}{20}\\i&=30\end{aligned}[/tex]

Thus the needed current should be 30 amperes.

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