Find an equation of variation for the given situation.The...

Find an equation of variation for the given situation.The volume $V$ of a given mass of a gas varies directly as the temperature $T$ and inversely as the pressure $P$. If $V=231 \mathrm{cm}^{3}$ when $T=42^{\circ}$ and $P=20 \mathrm{kg} / \mathrm{cm}^{2},$ what is the volume when $T=30^{\circ}$ and $P=15 \mathrm{kg} / \mathrm{cm}^{2} ?$

Key Concepts

Direct Variation

Direct variation refers to a relationship between two variables where an increase in one variable causes a proportional increase in the other. In such relationships, the two variables can be connected by a constant multiplier, making the relationship linear when one variable is plotted against the other.

Inverse Variation

Inverse variation describes a relationship where one variable increases as the other decreases such that their product remains constant. This means that an increase in one variable results in a proportional decrease in the other, establishing an inverse proportionality between them.

Constant of Proportionality

The constant of proportionality is the constant factor that relates two variables in both direct and inverse variation scenarios. It is determined by using known values from the relationship and is a key element in forming the equation that describes the variation.

Combined Variation

Combined variation involves a scenario where a variable depends directly on one factor and inversely on another. In this type of relationship, the variable can be expressed as being equal to a constant multiplied by one variable and divided by another, encapsulating both direct and inverse relationships in one formula.

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