Certainly! Let's break down each part of this problem step-by-step.
### (a) What is the decay rate of strontium-90?
The decay rate is given by the exponent in the decay function [tex]\( A(t) = A_0 e^{-kt} \)[/tex], where [tex]\( k \)[/tex] is the decay constant.
From the function [tex]\( A(t) = A_0 e^{-0.0244 t} \)[/tex], we can see that the decay rate [tex]\( k \)[/tex] is:
[tex]\[ k = 0.0244 \][/tex]
### (b) How much strontium-90 is left after 10 years?
To find the amount of strontium-90 left after 10 years, we use the given decay function:
[tex]\[ A(t) = A_0 e^{-kt} \][/tex]
Given:
- The initial amount [tex]\( A_0 = 500 \)[/tex] grams
- The decay rate [tex]\( k = 0.0244 \)[/tex]
- Time [tex]\( t = 10 \)[/tex] years
Substitute these values into the decay function:
[tex]\[ A(10) = 500 e^{-0.0244 \times 10} \][/tex]
When you compute this, the result is:
[tex]\[ A(10) \approx 391.74 \text{ grams} \][/tex]
### (c) When will only 200 grams of strontium-90 be left?
To find when only 200 grams will be left, we need to solve for [tex]\( t \)[/tex] in the decay equation:
[tex]\[ 200 = 500 e^{-0.0244 t} \][/tex]
First, divide both sides by 500:
[tex]\[ \frac{200}{500} = e^{-0.0244 t} \][/tex]
Simplify the fraction:
[tex]\[ 0.4 = e^{-0.0244 t} \][/tex]
Take the natural logarithm (ln) of both sides:
[tex]\[ \ln(0.4) = -0.0244 t \][/tex]
Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(0.4)}{-0.0244} \][/tex]
When you compute this, the result is:
[tex]\[ t \approx 37.55 \text{ years} \][/tex]
### (d) What is the half-life of strontium-90?
The half-life [tex]\( t_{1/2} \)[/tex] is the time it takes for half of the initial amount of the substance to decay. The half-life can be found using the formula:
[tex]\[ t_{1/2} = \frac{\ln(2)}{k} \][/tex]
Given the decay rate [tex]\( k = 0.0244 \)[/tex]:
[tex]\[ t_{1/2} = \frac{\ln(2)}{0.0244} \][/tex]
When you compute this, the result is:
[tex]\[ t_{1/2} \approx 28.41 \text{ years} \][/tex]
### Summary
(a) The decay rate [tex]\( k \)[/tex] is [tex]\( 0.0244 \)[/tex].
(b) The amount of strontium-90 left after 10 years is approximately 391.74 grams.
(c) It will take approximately 37.55 years for only 200 grams of strontium-90 to be left.
(d) The half-life of strontium-90 is approximately 28.41 years.
