#1. The pressure P of a compressed gas is inversely proportional to the volume V. If there is a pressure of 35 pounds per square inch when the volume of gas is 400 cubic inches, find the pressure when the gas is compressed to 100 square inches.
#2. A frictional force is necessary for a car to round a bend. The frictional force, F kilonewtons, varies directly as the square of the car's speed, V meters per second, and inversely as the radius of the bend, R meters. Write down a relationship between F, V and R.
#1. To solve this problem, we can use the concept of inverse proportionality. In an inverse relationship, when one variable increases, the other variable decreases, and vice versa. In this case, the pressure is inversely proportional to the volume of gas.
The formula for inverse proportionality is: P1 * V1 = P2 * V2
where P1 and V1 are the initial pressure and volume, and P2 and V2 are the new pressure and volume.
Given that the initial pressure is 35 pounds per square inch at a volume of 400 cubic inches, we can write the equation as: 35 * 400 = P2 * 100
Now, we can solve for P2 by rearranging the equation: P2 = (35 * 400) / 100
Calculating this expression, we find that P2 is equal to 140 pounds per square inch. Therefore, the pressure when the gas is compressed to 100 square inches is 140 pounds per square inch.
#2. To write down a relationship between F, V, and R, we can use the concepts of direct and inverse proportionality.
From the problem statement, we know that the frictional force, F, varies directly with the square of the car's speed, V, and inversely with the radius of the bend, R.
Mathematically, this can be expressed as: F = k * (V^2) / R
Here, k is a constant of proportionality, which reflects the specific relationship between the variables.
Thus, the relationship between F, V, and R is: F = k * (V^2) / R, where F is in kilonewtons, V is in meters per second, and R is in meters.