Question

Element X is a radioactive isotope such that its mass decreases by 59% every day. If an experiment starts out with 390 grams of Element X, write a function to represent the mass of the sample after tt days, where the hourly rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per hour, to the nearest hundredth of a percent.

Answer 1

Answer:

The function that represents the mass of the sample after t days is $A(t) = 390(0.41)^t$.

The percentage rate of change per hour is of -2.46%.

Step-by-step explanation:

Exponential amount of decay:

The exponential amount of decay for an amount of a substance after t days is given by:

$A(t) = A(0)(1-r)^t$

In which A(0) is the initial amount, and r is the decay rate, as a decimal.

Element X is a radioactive isotope such that its mass decreases by 59% every day. The experiments starts out with 390 grams of Element X.

This means, respectively, that $r = 0.59, A(0) = 390$

So

$A(t) = A(0)(1-r)^t$

$A(t) = 390(1-0.59)^t$

$A(t) = 390(0.41)^t$

The function that represents the mass of the sample after t days is $A(t) = 390(0.41)^t$.

Hourly rate of change:

Decreases by 59% every day, which means that for 24 hours, the rate of change is of -59%. So

-59%/24 = -2.46%

The percentage rate of change per hour is of -2.46%.