Question

Element X is a radioactive isotope such that its mass decreases by 26% every day. If an experiment starts out with 810 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 hourly decay rate is of 1.25%, so the hourly rate of change is of -1.25%.

The function to represent the mass of the sample after t days is $A(t) = 810(0.74)^t$

Step-by-step explanation:

Exponential equation of decay:

The exponential equation for the amount of a substance 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.

Hourly rate of change:

Decreases 26% by day. A day has 24 hours. This means that $A(24) = (1-0.26)A(0) = 0.74A(0)$; We use this to find r.

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

$0.74A(0) = A(0)(1-r)^{24}$

$(1-r)^{24} = 0.74$

$\sqrt[24]{(1-r)^{24}} = \sqrt[24]{0.74}$

$1 - r = (0.74)^{\frac{1}{24}}$

$1 - r = 0.9875$

$r = 1 - 0.9875 = 0.0125$

The hourly decay rate is of 1.25%, so the hourly rate of change is of -1.25%.

Starts out with 810 grams of Element X

This means that $A(0) = 810$

Element X is a radioactive isotope such that its mass decreases by 26% every day.

This means that we use, for this equation, r = 0.26.

The equation is:

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

$A(t) = 810(1 - 0.26)^t$

$A(t) = 810(0.74)^t$

The function to represent the mass of the sample after t days is $A(t) = 810(0.74)^t$