Help please with math problem

Question

A of a radioactive substance has an initial mass of 981.5mg This substance follows a continuous exponential decay model and has a half-life of 16 hours. Let (t) be the time in hours) since the start of the experiment , and let (y) be the amount of the substance at time (t). Write a formula relating y to t. Use exact expressions to in the missing parts of the formula Do not use approximations. How much will be present in 22 hours? Do not round any intermediate computations , and round your answer to the nearest tenth

Answer 1

We are given that the substance follows a continuous exponential decay model. That means the same portion of the remaining substance decays per time.

This alone means that f(t) = A * r^t, where A is the initial mass in mg and r is the proportion of the substance that is left (does not decay) per hour.

The question is asking for f(22).

We are given A=981.5, but we need to figure out r.

Understanding f(t) = A * r^t:

If zero hours have passed, then t=0 and r^t = r^0 = 1. f(0) = A.

If three hours have passed, then t=3 and f(t) = A * r * r * r. E.g. "It decays for 1 hour 3 times".

If 16 hours have passed, f(t) = A * r^16.

First-time way of looking at it:

We are given that the half-life is 16 hours.

This literally means that f(t+16) = (1/2) * f(t).

(For example, f(16) = (1/2) * f(0) = 1/2 * 981.5 = 490.75).

Putting it together:

f(t+16) = A * r^(t+16) = A * r^t * r^16

f(t+16) = (1/2) * f(t) = (1/2) * A * r^t.

So A * r^t * r^16 = (1/2) * A * r^t.

That means r^16 = (1/2)! E.g. "It takes 1 hour of decay to happen 16 times in order for only half the substance to remain. "

That means r = (1/2)^(1/16).

f(t) = A * r^t = 981.5 * [(1/2)^(1/16)]^t

Plug in t=22 to figure out f(22) and don't forget to round the final answer to the nearest tenth.

Simpler way of looking at it:



f(t₁₆) = A * r₁₆^t₁₆, where t₁₆ is the number of 16-hour chunks that have passed by.

Since the half-life is 16 hours, we get that r₁₆=0.5.

f(t₁₆) = 981.5 * 0.5 ^ t₁₆.

For 22 hours, t₁₆ = 22/16. That's how many 16-hour chunks have passed by.

Plug in 22/16 to get the answer: f(22/16) = 981.5 * 0.5 ^ (22/16).

Answer 2

We can write the exponential decay as y = 981.5 (1/2)^t/16 -- this yields 1/2 of the starting amount after 16 hours, 1/4 after 32 hours, etc. For t = 22 hours, y = 981.5 (0.5)^22/16 = 981.5 (0.38555) = 378.4 mg.

Answer 3