For each ($P, V$) pair, type the pressure in the x column and the volume in the y-column. Then click 'Resize window to fit data.'
Choose the power regression option. Copy the equation, using three significant figures, to match the data.
[tex]\begin{array}{l}<
V=51.4 \quad \checkmark \
P^{\wedge}-1.00 \quad X \Rightarrow-0.999
\end{array}[/tex]
Notice that the exponent is very close to -1. The equation is essentially in the form of
[tex]V=k P^{-1}[/tex]
Is this a proportion, an inverse proportion, a linear relationship, a quadratic relationship, or a direct proportion?
Answer (2)
The relationship between volume (V) and pressure (P) given by the power regression equation is an inverse proportion, expressed as V = k P^{-1}. This means that as pressure increases, the volume decreases, maintaining a constant product. This relationship is key in understanding gas behavior, such as in Boyle's Law.
;
The problem provides a power regression equation Va pp ro x 51.387 P − 0.999 .
The exponent is approximately -1, so the equation is V = k P − 1 .
This equation represents an inverse proportion between V and P.
Therefore, the relationship is an inverse proportion .
Explanation
Understanding the Problem The problem provides a power regression equation relating pressure (P) and volume (V) as y a pp ro x 51.387 x − 0.999 , where x represents pressure and y represents volume. The exponent is very close to -1, suggesting an inverse relationship. We need to identify the correct relationship between V and P.
Rewriting the Equation The power regression equation can be rewritten as V = k P − 1 , where k is a constant. This is because in the given equation, y corresponds to V , x corresponds to P , and the constant 51.387 corresponds to k , while the exponent − 0.999 is approximately − 1 .
Identifying the Relationship The equation V = k P − 1 can also be written as V = P k or V P = k . This form clearly indicates that as pressure (P) increases, volume (V) decreases proportionally, and vice versa, maintaining a constant product. This is the definition of an inverse proportion.
Conclusion Therefore, the relationship between volume (V) and pressure (P) is an inverse proportion.
Examples
In real life, the relationship between pressure and volume is commonly observed in scenarios involving gases, such as in engines or balloons. For example, if you compress a gas (increase the pressure), its volume decreases, assuming the temperature remains constant. This principle is described by Boyle's Law, which states that for a fixed amount of gas at a constant temperature, the pressure and volume are inversely proportional. Mathematically, this is expressed as P 1 V 1 = P 2 V 2 , where P 1 and V 1 are the initial pressure and volume, and P 2 and V 2 are the final pressure and volume. This concept is crucial in understanding how engines work and how gases behave under different conditions.
