Suppose that a radioactive substance decays with a half-life of years. Find a function that expresses the amount of substance present at time t if there are g of the substance initially present. Round answers to 3 decimal places if need be.

Question

Suppose that a radioactive substance decays with a half-life of  years. Find a function that expresses the amount of substance present at time t if there are  g of the substance initially present. Round answers to 3 decimal places if need be.

Quizplus · Accepted Answer

The amount of substance present after time t can be expressed as: 
A(t) = A0(1/2)^(t/T), 
where A0 is the initial amount of substance present and T is the half-life of the substance. 
In this case, A0 = 11ead0bc_8bf2_41ad_99a0_75b50246cf71_TB6988_11 (given) and T = 11ead0bc_8bf2_1a9b_99a0_1fd1d9b48790_TB6988_11 (given). 
Substituting these values, we get: 
A(t) = 11ead0bc_8bf2_41ad_99a0_75b50246cf71_TB6988_11 * (1/2)^(t/11ead0bc_8bf2_1a9b_99a0_1fd1d9b48790_TB6988_11) 
Taking logarithms on both sides, we can simplify this expression to: 
log(A(t)) = log(11ead0bc_8bf2_41ad_99a0_75b50246cf71_TB6988_11) - (t/11ead0bc_8bf2_1a9b_99a0_1fd1d9b48790_TB6988_11) * log(2) 
This is in the form y = mx + b, where y = log(A(t)), x = t, m = -(log(2))/11ead0bc_8bf2_1a9b_99a0_1fd1d9b48790_TB6988_11, and b = log(11ead0bc_8bf2_41ad_99a0_75b50246cf71_TB6988_11). 
Therefore, the graph of log(A(t)) vs t is a straight line with slope -(log(2))/11ead0bc_8bf2_1a9b_99a0_1fd1d9b48790_TB6988_11 and y-intercept log(11ead0bc_8bf2_41ad_99a0_75b50246cf71_TB6988_11). 
Using the given values of t, we can calculate the amount of substance present at each time point and choose the option that matches the calculated value. 
For t = 11ead0bc_8bf2_1419_99a0_91aa1571c19f_TB6988_11, we get:
A(t) = 11ead0bc_8bf2_41ad_99a0_75b50246cf71_TB6988_11 * (1/2)^(11ead0bc_8bf2_1419_99a0_91aa1571c19f_TB6988_11/11ead0bc_8bf2_1a9b_99a0_1fd1d9b48790_TB6988_11) 
= 11ead0bc_8bf2_41ad_99a0_75b50246cf71_TB6988_11 * 0.5 
= 11ead0bc_8bf2_68be_99a0_bbbf96c3a0b7_TB6988_11 
So, the correct answer is D.

Suppose That a Radioactive Substance Decays with a Half-Life of