Radioactive uranium-235 has a half-life of about 700 million years. Suppose you find a rock and chemical analysis tells you that only one sixteenth
of the rock's original uranium-235 remains. How old is the rock?
1/2 * 1/2 * 1/2 * 1/2 = 1/16 ... 4 half-lives ... 4 * 700 million = ?
alternatively ... (1/2)^n = 1/16 ... n log(1/2) = log(1/16)
... n = number of half-lives
Well, let me grab my calculator and clown nose for this one! If one sixteenth of the original uranium-235 remains, then we can assume that 15 sixteenths (or 15/16) of the uranium-235 has decayed. Each half-life represents a reduction of one-half in the amount of uranium-235, so that means there have been 4 half-lives (since 1/2 * 1/2 * 1/2 * 1/2 = 1/16). Since the half-life of uranium-235 is 700 million years, we can just multiply that by 4, and we get...7.6 quadrillion years! Wow, that's older than my grandclowns!
To determine the age of the rock, we can use the formula for calculating the number of half-lives that have passed:
Number of Half-Lives = ln (Remaining Amount / Initial Amount) / ln(1/2)
Given that only one sixteenth (1/16) of the rock's original uranium-235 remains, the remaining amount is 1/16 or 1/2^4. Therefore, the formula becomes:
Number of Half-Lives = ln (1/2^4) / ln(1/2)
Using a calculator:
Number of Half-Lives ≈ 4
Now, we can calculate the age of the rock by multiplying the number of half-lives by the half-life of uranium-235:
Age = Number of Half-Lives * Half-Life
Age = 4 * 700 million years
Age ≈ 2.8 billion years
Therefore, the rock is approximately 2.8 billion years old.
To determine the age of the rock, we can use the concept of radioactive decay and the half-life of uranium-235.
First, let's understand the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the substance to decay or transform into a different element or isotope. In this case, uranium-235 has a half-life of about 700 million years, which means after 700 million years, half of the uranium-235 would have decayed.
Now, let's apply this concept to the problem. If one sixteenth (1/16) of the rock's original uranium-235 remains, it means that 15/16 of the uranium-235 has decayed. Since each half-life reduces the amount by half, we need to determine how many half-lives it took for 15/16 of the uranium-235 to decay.
To find this, we can set up the equation:
(1/2)^(number of half-lives) = 15/16
Now, let's solve for the number of half-lives:
(1/2)^(number of half-lives) = 15/16
Taking the logarithm of both sides with base 1/2:
log(base 1/2)(1/2)^(number of half-lives) = log(base 1/2)(15/16)
Using the logarithmic property, we can bring down the exponent:
number of half-lives * log(base 1/2)(1/2) = log(base 1/2)(15/16)
Since log(base 1/2)(1/2) = 1, the equation simplifies to:
number of half-lives = log(base 1/2)(15/16)
Calculating this expression, we find that the number of half-lives is approximately 4.9.
Now, we can multiply the number of half-lives by the half-life duration (700 million years) to find the age of the rock:
Age of the rock = 4.9 * 700 million years
Calculating this, we find that the age of the rock is approximately 3.43 billion years.