Write the statement as a power function equation. Use k for constant of variation if one is not given.
a. Charles's law states the volume V of an enclosed ideal gas at a constant pressure varies directly as the absolute temperature T.
b. The volume V of a circular cylinder with a fixed height is proportional to the square of its radius.
c. the current I in an electrical circuit is inversely proportional to the resistance R, with constant of variation V.
I can't figure out how to do this. I was looking at my old notes but can't seem to come up with something.
No problem! I'd be happy to help you write these statements as power function equations. Let's break it down:
a. Charles's law states the volume V of an enclosed ideal gas at a constant pressure varies directly as the absolute temperature T.
In a power function equation, the general form is y = kx^n, where y represents the dependent variable, x represents the independent variable, k is the constant of variation, and n is the exponent.
In this case, we are given that the volume V varies directly with the absolute temperature T. "Varies directly" means that the exponent, n, is equal to 1. The constant of variation, k, will be represented by another letter since it is not given explicitly in the statement. Let's use c for this constant.
Therefore, the power function equation for Charles's law can be written as:
V = cT^1
Since any number raised to the power of 1 remains unchanged, we can simplify the equation to:
V = cT
b. The volume V of a circular cylinder with a fixed height is proportional to the square of its radius.
Similar to before, in a power function equation, we have y = kx^n. In this case, the volume V is proportional to the square of the radius, which means the exponent n is 2.
Let's denote the constant of variation by k, since it is not explicitly given.
Therefore, the power function equation can be written as:
V = kr^2
c. The current I in an electrical circuit is inversely proportional to the resistance R with a constant of variation V.
When two variables are inversely proportional, their product remains constant. In this case, the current I and resistance R are inversely proportional, so we can write the equation as:
I * R = V
Here, V represents the constant of variation.
I hope this helps you understand how to write power function equations for these situations! If you have any more questions, feel free to ask.