The half-life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to half its initial mass. Starting with 195 grams of a radioactive isotope, how much will be left after 4 half-lives? Round your answer to the nearest gram.

Question

anonymous · Answer

@whpalmer4

whpalmer4 · Answer

After 1 half life, you have 1/2 the starting amount.  After another half life, you have 1/2 of 1/2 of the starting amount.  Continue the pattern...

anonymous · Answer

Would it be 12.187??

whpalmer4 · Answer

yep, 12.1875

anonymous · Answer

How would you round that to the nearest gram?

whpalmer4 · Answer

now, it's not that the material that decayed just disappeared, it's probably still there, just a different element now.  Sometimes the decay does transmute a solid element into a gas, however, and then that gas often decays back into a solid.  Radium, a metal, for example, decays by emitting an alpha particle and becoming radon, a gas, which then decays by emitting another alpha particle and becoming polonium, a metal.  Result is that you get little itty bitty specks of polonium sticking to dust particles in the air, which is really bad news when you inhale those dust particles!  Alpha particles are stopped by a sheet of paper or skin, but if you inhale something that emits alpha particles, they can do a lot of damage internally.

whpalmer4 · Answer

well, the nearest gram is either 12 grams, or 13 grams.  Which one is closer to 12.1875 grams?

whpalmer4 · Answer

In general, to round to a given decimal place, you add 1/2 of the value of that decimal place, and then drop any digits to the right (or make them 0 if you are to the left of the decimal point).

123 rounded to the nearest 10:  
123 + 10/2 = 123 + 5 = 128, we are rounding to the nearest 10 so that becomes 120.

128 rounded to the nearest 10:
128 + 10/2 = 128 + 5 = 133, we are rounding to the nearest 10 so that becomes 130.

1.943 rounded to the nearest 0.01:
1.943 + 0.01/2 = 1.943 + 0.005 = 1.948, rounding to the nearest 0.01 so we drop the digits to the right of that decimal place giving us 1.94.

anonymous · Answer

I only need help with two more questions. I did half of the work, do you think you could tell me if i'm doing it right? @whpalmer4

whpalmer4 · Answer

sure, what do you have?

anonymous · Answer

The first one says:
Solve the quadratic equation by completing the square. x^2-8x+1=0

First, choose the appropriate form and fill in the blanks with the correct numbers. 
Then, solve the equation. Round your answer to the nearest hundredth. 
If there is more than one solution, separate them with commas.

FORMS:
A. (x+?)^2 = ?
B. (x-?)^2 = ?

x=?

anonymous · Answer

For my answer I put A. (x+4)^2 = 15

x=7.87 and x=-0.13
@whpalmer4

whpalmer4 · Answer

You made a sign error, want to try again?

anonymous · Answer

-7.87 and 0.13?

whpalmer4 · Answer

You are solving $$x^2-8x+1=0$$right?

anonymous · Answer

Yes.

whpalmer4 · Answer

1 is pretty close to 0.  If our equation was $$x^2 - 8x + 0 = 0$$a solution would be $x = 8$ right?  That means that one our solutions ought to be pretty close to $8$.  You had $x = 7.87$ the first time around, and that looks plausible.  We could even plug it in to check it:
$$(7.87)^2 - 8(7.87) +1 = 0$$$$61.9-62.96 + 1 \approx 0$$
We rounded, and there's an irrational number in there, so it didn't come exactly.  NBD.

But the other solution you had was $x=-0.13$:

$$(-0.13)^2 -8(-0.13) + 1 = 0$$The first term is going to be about 0.01, second term is going to be about 1, so that gives us $$0.1+1+1 \ne 0 $$

So, what was the mistake?

anonymous · Answer

The negative sign was a mistake.

whpalmer4 · Answer

Yes, 0.127017 would be much closer :-)

whpalmer4 · Answer

but you mentioned above "for my answer, I put A: (x+4)^2 = 15"

and if we have a $-8x$ term in our equation, it's going to be $(x-4)^2$, right?

anonymous · Answer

Ah, I got it. messed up the signs again.

anonymous · Answer

For the second problem it's the same thing except the equation is x^2−18x+79=0

So my answer for the first part would be B. (x-9)^2 = 2
And x = 10.41 and x=7.6 
@whpalmer4

whpalmer4 · Answer

yes, those ar correct

anonymous · Answer

My test said it was incorrect.

anonymous · Answer

Oh wait, I know what I did wrong! I rounded incorrectly.

whpalmer4 · Answer

yeah, sorry, I didn't notice what it said about rounding and so didn't check that.