Question

Before kerosene can be used as jet fuel, federal regulations require that the pollutants in it be removed by passing the kerosene through clay. Suppose that the clay is in a pipe and that each foot of the pipe will remove 20% of the pollutants that enter it.

a. If P0 is the initial number of pollutant and P (x) represents the quantity of pollutants left after x feet pipe, derive an expression for P (x) = Po(n), and assume that P0 = 1.

Answer 1

The quantity of pollutants left after x feet of clay pipe is given by the equation P(x) = P0 × 0.8^x, assuming 20% is removed per foot of pipe. If P0 equals 1, the equation simplifies to P(x) = 0.8^x.

Step-by-step explanation:

To determine the quantity of pollutants left after kerosene has passed through x feet of clay pipe, we must take into account that each foot removes 20% of the pollutants entering it. The equation for this exponential decay process can be derived as follows:

Since 20% of the pollutants are removed per foot of pipe, 80% remain. After one foot of pipe, the quantity of pollutants left is 0.8 of the initial quantity P0. After two feet, it is 0.8 squared of the initial quantity, and so on. Therefore, for x feet of pipe, the quantity of pollutants left is 0.8 to the power of x of the initial quantity.

The general expression for P(x) is therefore P(x) = P0 × 0.8x, where P0 is the initial amount of pollutants, and x is the number of feet of pipe the kerosene has passed through.

If we assume that the initial number of pollutants P0 is 1, the equation simplifies to P(x) = 0.8x.