assume the substance has a half-life of 11 years and the initial amount is 126 grams.How long will it be until only 15 % remains?

To determine how long it will take for only 15% of the substance to remain, we need to use the concept of exponential decay.

First, let's define the equation for exponential decay:

A = A₀ * (1/2)^(t/h)

Where:
A₀ is the initial amount of the substance,
A is the remaining amount of the substance after time t,
h is the half-life of the substance,
t is the time elapsed.

In this case, we're given that the initial amount is 126 grams (A₀ = 126 grams), and we want to find the time (t) required for 15% (0.15) of the substance to remain (A = 0.15 * A₀).

Plug in the values into the equation:

0.15 * A₀ = A₀ * (1/2)^(t/11)

Now, we can cancel out A₀ on both sides:

0.15 = (1/2)^(t/11)

Taking the logarithm of both sides (base 2 since we have 1/2):

log₂(0.15) = log₂((1/2)^(t/11))

Using the logarithmic property (logₐ(b^c) = c * logₐ(b)):

log₂(0.15) = (t/11) * log₂(1/2)

Now, solve for t by isolating it:

t/11 = log₂(0.15) / log₂(1/2)

Simplify the right side using the property log₂(b) = log(c)/log(b):

t/11 = log(0.15) / log(1/2)

Finally, solve for t:

t = 11 * (log(0.15) / log(1/2))

Using a calculator, we find that t ≈ 30.69 years.

Therefore, it will take approximately 30.69 years for only 15% of the substance to remain.

To determine how long it will take for only 15% of the substance to remain, we need to calculate the number of half-lives it will take for the amount to decrease to 15% of the initial amount.

Step 1: Calculate the amount of the substance that remains after each half-life.

After the first half-life: Remaining amount = Initial amount * (1/2) = 126 grams * (1/2) = 63 grams

After the second half-life: Remaining amount = Previous remaining amount * (1/2) = 63 grams * (1/2) = 31.5 grams

After the third half-life: Remaining amount = Previous remaining amount * (1/2) = 31.5 grams * (1/2) = 15.75 grams

Step 2: Calculate the percentage of the initial amount that remains after the number of half-lives.

Percentage remaining = (Remaining amount / Initial amount) * 100

After three half-lives: Percentage remaining = (15.75 grams / 126 grams) * 100 = 12.5%

Step 3: Determine the number of additional half-lives needed to reach 15% remaining.

Let's assume x is the number of additional half-lives needed.

Percentage remaining after x additional half-lives = (15/100) * 126 grams

Setting up the equation:

(1/2)^x * 126 grams = (15/100) * 126 grams

Dividing both sides by 126 grams:

(1/2)^x = 15/100

Taking the logarithm (base 2) of both sides:

log2((1/2)^x) = log2(15/100)

Using the logarithmic identity: log_b(a^c) = c * log_b(a)

x * log2(1/2) = log2(15/100)

Since log2(1/2) is equal to -1:

-x = log2(15/100)

Dividing both sides by -1:

x = -log2(15/100)

Using a calculator:

x ≈ 3.91 (rounded to two decimal places)

Therefore, it will take approximately 3.91 additional half-lives for only 15% of the initial amount (126 grams) to remain.

1/2 = e^k(11)

ln .5 = 11 k
11 k = -.5978
k = -.05435
so
.15 = e^-.05435 t
-1.897 = - .05435 t
t = 34.9 years