Question

Use a variation model to solve for the unknown value.
The current in a wire varies directly as the voltage and inversely as the resistance. If the current is 15 amperes (A) when the voltage is 80 volts (V) and the resistance is 8 ohms (52), find the current when the voltage is 111 V and the resistance is 12 $.

Answer 1

To solve the problem, we can use the formula for current in a wire, which is given by Ohm's law:

\[
I = \frac{V}{R}
\]

However, since the current varies directly as the voltage and inversely as the resistance, we can write this relationship as:

\[
I = k \cdot \frac{V}{R}
\]

where \( I \) is the current in amperes, \( V \) is the voltage in volts, \( R \) is the resistance in ohms, and \( k \) is a constant of proportionality.

From the problem, we know the following values:

- \( I = 15 \) A when \( V = 80 \) V and \( R = 8 \) ohms.

We can first calculate the constant \( k \):

\[
15 = k \cdot \frac{80}{8}
\]

Simplifying the fraction:

\[
\frac{80}{8} = 10
\]

So we have:

\[
15 = k \cdot 10
\]

Now, solving for \( k \):

\[
k = \frac{15}{10} = 1.5
\]

Now we have established that the relationship between current, voltage, and resistance can be expressed as:

\[
I = 1.5 \cdot \frac{V}{R}
\]

Next, we need to find the current \( I \) when \( V = 111 \) V and \( R = 12 \) ohms:

\[
I = 1.5 \cdot \frac{111}{12}
\]

Calculating \( \frac{111}{12} \):

\[
\frac{111}{12} = 9.25
\]

Now substituting back into the formula for \( I \):

\[
I = 1.5 \cdot 9.25
\]

Calculating \( 1.5 \cdot 9.25 \):

\[
I = 13.875
\]

Therefore, the current when the voltage is 111 V and the resistance is 12 ohms is

\[
\boxed{13.875 \text{ A}}
\]