The mass of a radioactive sample is represented in the graph below. The initial mass of 32 mg decays to 8 mg after 21 hours.
1)What is the half-life of the radioactive sample, in minutes?
2)Solve each equation.
a. 4 8x-1 = 8
b. 3^(2x-5) = 1/27x
8 is 1/4 of 32, so two half-lives = 21 hours.
half-life is 10.5 hours
4^(8x-1) = (2^2)^(8x-1) = 2^(16x-2) = 8 = 2^3
so, 16x-2 = 3
x = 5/16
assuming 1/27x = (1/27)*x,
3^(2x-5) = (3^-3)x
(2x-5)log3 = -3log3 + logx
(2x-2)log3 = logx
nope
assuming a typo, and the x on the right does not belong,
3^(2x-5) = 1/27 = 3^-3, so
2x-5 = -3
x = 1
To answer the first question regarding the half-life of the radioactive sample, we need to understand the concept of half-life. The half-life refers to the time it takes for half of a radioactive substance to decay.
Given that the initial mass of 32 mg decays to 8 mg after 21 hours, we know that the sample has lost half of its mass. This means that the half-life corresponds to the time it took for this decay to occur.
To determine the half-life in minutes, we need to convert the given time of 21 hours into minutes. Since there are 60 minutes in an hour, the conversion will be as follows:
21 hours x 60 minutes/hour = 1260 minutes
Therefore, the half-life of the radioactive sample is 1260 minutes.
Moving on to the second part of your question, where we need to solve equations:
a. 4(8x-1) = 8
To solve this equation, we can begin by distributing the 4 to the terms inside parentheses:
32x - 4 = 8
Next, we can isolate the variable x by moving the constant term (4) to the other side of the equation:
32x = 8 + 4
32x = 12
Finally, we solve for x by dividing both sides of the equation by 32:
x = 12/32
x = 0.375
Therefore, the value of x that solves the equation is 0.375.
b. 3^(2x-5) = 1/27x
To solve this equation, we can start by getting rid of the fraction on the right side. We know that 1/27x can be rewritten as (1/3^3)^x, which simplifies to 3^(-3x):
3^(2x-5) = 3^(-3x)
Since the bases on both sides of the equation are the same (3), we can equate the exponents:
2x - 5 = -3x
Next, let's isolate x by moving the variable terms to one side of the equation and the constant term to the other side:
2x + 3x = 5
Simplifying further:
5x = 5
Finally, we solve for x by dividing both sides by 5:
x = 5/5
x = 1
Therefore, the value of x that solves the equation is 1.