URGENT help!

3. A rock sample was dated using potassium-40, Neasyrenebt ubducated tgat 1/8 of the original parent isotope is left in the rock sample. How old is the rock?

5. When a sample of lava solidfied, it contained 28g of uranium-238. It that lava sample was later found to contain only 7g of uranium-238, how many years had passed since the lava solidified?

6. AFter 25 years, the number of radioactive cobalt atoms in a sample is reduced to 1/32 of the original count. What is the half-life of this isotope?

How would you find the answers to these questions?
TEll me the steps.

Thank you very much.

#3 has so many typos I can't translate it. Please repost.

#5. ln(No/N) = kt. I don't see k listed in the problem but it can be calculated from k = 0.693/t1/2. Thus, look up the half-life for U-238, substitute in the second equation to find k, then solve the first equation for t. No = 28 g. N = 7 g. The unit for t will be th same units used for the half-life.

#6. Use ln(No/N) = kt.I would make up a number (some convenient number such as 160) for No, then 1/32 x 160 will be N and you can solve for k since you are given t. Then k = 0.693/t1/2 and solve for t1/2.

To find the answers to these questions, you can use the concept of radioactive decay and the formula for calculating the remaining amount of radioactive material.

Before diving into the specific steps for each question, let's understand the general formula for radioactive decay:

N(t) = N(0) * (1/2)^(t / T)

Where:
- N(t) is the amount of remaining radioactive material at time t
- N(0) is the initial amount of radioactive material
- t is the time that has passed
- T is the half-life of the isotope

Now, let's go through each question step by step:

3. A rock sample was dated using potassium-40, and you're told that 1/8 of the original parent isotope is left in the rock sample. To find out how old the rock is, you need to determine how many half-lives have passed.

Step 1: Set up the equation using the formula for radioactive decay:
1/8 = (1/2)^(t / T)

Step 2: Solve for t by taking the logarithm of both sides of the equation:
log(1/8) = (t / T) * log(1/2)

Step 3: Simplify the equation and solve for t:
-3 = (t / T) * -1
3 = t / T

So, the rock is approximately three times the half-life of potassium-40 old.

5. When a sample of lava solidified, it contained 28g of uranium-238. Now, it contains only 7g of uranium-238. To determine the time that has passed, we'll use the same radioactive decay formula but solve for T instead of t.

Step 1: Set up the equation using the formula for radioactive decay:
7g = 28g * (1/2)^(t / T)

Step 2: Simplify the equation by dividing both sides by 28g:
1/4 = (1/2)^(t / T)

Step 3: Take the logarithm of both sides of the equation:
log(1/4) = (t / T) * log(1/2)

Step 4: Simplify the equation and solve for T:
-2 = (t / T) * -1
2 = t / T

So, the time that has passed since the lava solidified is approximately two half-lives of uranium-238.

6. After 25 years, the number of radioactive cobalt atoms in a sample is reduced to 1/32 of the original count. To find the half-life of this isotope, we'll follow similar steps as in the previous questions.

Step 1: Set up the equation using the formula for radioactive decay:
1/32 = (1/2)^(25 / T)

Step 2: Take the logarithm of both sides of the equation:
log(1/32) = (25 / T) * log(1/2)

Step 3: Simplify the equation and solve for T:
-5 = (25 / T) * -1
5 = 25 / T

So, the half-life of the radioactive cobalt is approximately five years.

Remember, these calculations are based on assumptions and simplifications. In practice, it is essential to consider other factors and perform more precise measurements for accurate results.