assume the half-life of the substance is 31 days and the initial
amount is 183.9 grams.
(a) Fill in the right hand side of the following equation which expresses
the amount A of the substance as a function of time t (the coefficient of t in the exponent should have at least four significant digits):
(b) When will the substance be reduced to 2.4 grams?
(use decimals.)
the second answer is 194.052 days
There is not "followiong equation" to fill in.
Amount left = 183.9*2^(-t/31)
You don't need 4 sig figs in the exponent in this form if the half life is exactly 31 days.
(b) When amount left = 2.4,
2.4/183.9 = 0.01088 = 2^(-t/31)
-4.5212 = (-t/31)*ln 2 = -0.02236t
t = 202.2 days
(a) The general equation for the decay of a substance with a half-life is given by:
A = A₀ * (1/2)^(t / t₁/₂)
where:
A is the amount of the substance at time t,
A₀ is the initial amount of the substance,
t is the time, and
t₁/₂ is the half-life of the substance.
In this case, the half-life (t₁/₂) is 31 days and the initial amount (A₀) is 183.9 grams.
So, the equation representing the amount A of the substance as a function of time t becomes:
A = 183.9 * (1/2)^(t / 31)
(b) To find when the substance will be reduced to 2.4 grams, we can use the equation from part (a) and solve for t.
2.4 = 183.9 * (1/2)^(t / 31)
To solve for t, we can take the logarithm (base 1/2) of both sides:
log(2.4) = log(183.9 * (1/2)^(t / 31))
Using logarithmic properties, we can simplify the equation:
log(2.4) = log(183.9) + (t / 31) * log(1/2)
Now, solve for t:
(t / 31) = (log(2.4) - log(183.9)) / log(1/2)
t = 31 * ((log(2.4) - log(183.9)) / log(1/2))
Using a calculator, we can evaluate the right-hand side to find the value of t.
(a) The equation that expresses the amount A of the substance as a function of time t is given by:
A = A₀ * (1/2)^(t / t₁/₂)
Where:
A = Amount of substance at time t
A₀ = Initial amount of substance
t = Time
t₁/₂ = Half-life of the substance
In this case, the initial amount A₀ is 183.9 grams and the half-life t₁/₂ is 31 days. So the equation becomes:
A = 183.9 * (1/2)^(t / 31)
Note: To calculate the exponent, t / 31, use the ratio of time t to the half-life t₁/₂.
(b) To find when the substance will be reduced to 2.4 grams, we need to solve the equation using the given amount A = 2.4 grams:
2.4 = 183.9 * (1/2)^(t / 31)
To solve for t, we need to isolate the variable t. Divide both sides of the equation by 183.9:
2.4 / 183.9 = (1/2)^(t / 31)
Now, take the logarithm (base 1/2) of both sides of the equation:
log₁/₂(2.4 / 183.9) = t / 31
Using the change of base formula, we can convert the logarithm to a base 10 logarithm:
log₁/₂(2.4 / 183.9) = log₁₀(2.4 / 183.9) / log₁₀(1/2)
Now, calculate the value of the right-hand side of the equation using a calculator or software:
log₁₀(2.4 / 183.9) / log₁₀(1/2) ≈ -4.116
Finally, multiply both sides of the equation by 31 to isolate the variable t:
t = -4.116 * 31 ≈ -127.65
Since time cannot be negative, we take the absolute value of t:
|t| ≈ |-127.65| ≈ 127.65
Therefore, the substance will be reduced to 2.4 grams approximately 127.65 days after the initial amount.