The radioactive element polonium decays according to the law given below where Q0 is the initial amount and the time t is measured in days.
Q(t) = Q0 · 2-(t/140)
If the amount of polonium left after 700 days is 45 mg, what was the initial amount present?
mg
Q0 2^-(700/140) = 45
Note that 700 days is 5 half-lives
So, 1/32 of Q0 is left
Q0 = 32*45 = 1440
Well, well, well, looks like polonium is at it again, decaying and causing trouble. So, you're telling me that after 700 days, we're left with a mere 45 mg of this radioactive element? Oh boy, that's not good news.
But fear not, my curious friend, because I've got just the formula for this situation. According to the radioactive decay law you provided, Q(t) = Q0 · 2^(-t/140). Funny little function, isn't it?
Now, let's plug in the values we know. After 700 days, Q(700) = 45 mg. So we've got, 45 = Q0 · 2^(-700/140). Hang on, let me put on my mathematician hat and do some calculations for you.
Divide both sides by Q0 and we get 45/Q0 = 2^(-700/140). Now, let's bring the power of humor into the equation. Since 2^(10/2) = 2^5, we can rewrite the equation as 45/Q0 = 2^(-5). Are you still with me?
Now comes the fun part. Flip both sides of the equation, and we get 1/(45/Q0) = 1/(2^(-5)). Are you ready? This simplifies down to Q0/45 = 2^5.
Now, don't lose your sense of humor, my friend. Multiply both sides by 45, and we get Q0 = 45 · 2^5. Simplify that, and you'll find Q0 = 1440 mg. Tada!
So, the initial amount of polonium present was 1440 mg. Isn't it amazing how humor and math can go hand in hand?
To find the initial amount of polonium present (Q0), we can plug in the given values into the decay formula and solve for Q0.
The decay formula for polonium is:
Q(t) = Q0 · 2^(-t/140)
We know that the amount of polonium left after 700 days is 45 mg, so we can set up the equation as follows:
45 = Q0 · 2^(-700/140)
Simplifying the equation further:
45 = Q0 · 2^(-5)
Since 2^(-5) equals 1/32, we can rewrite the equation as:
45 = Q0/32
To isolate Q0, we can multiply both sides of the equation by 32:
32 * 45 = Q0
So, the initial amount present (Q0) is 1,440 mg.
To find the initial amount of polonium present, we can use the given information and the decay formula for polonium.
The decay formula is given as:
Q(t) = Q0 · 2^(-t/140)
We are given that the amount of polonium left after 700 days is 45 mg. Using this information, we can substitute the values into the formula and solve for Q0.
45 = Q0 · 2^(-700/140)
To solve for Q0, we need to isolate it on one side of the equation.
Dividing both sides of the equation by 2^(-700/140), we get:
45 / 2^(-700/140) = Q0
Simplifying the right side of the equation, we have:
45 / 2^(-5) = Q0
Using the properties of exponents, 2^(-5) is equal to 1/2^5 or 1/32. So we have:
45 / (1/32) = Q0
Simplifying further, we get:
45 * 32 = Q0
1440 = Q0
Therefore, the initial amount of polonium present was 1440 mg.