half-life: assume quantity has 5.2% decrease in 7.3 hours; find half-life

[ellieklinger]

“assume a quantity decreases by 5.2% in 7.3 hours. What is the half-life of the substance?”

i’m confused about where to start with this problem or what formula to solve it

 

[The Highlander]

“assume a quantity decreases by 5.2% in 7.3 hours. What is the half-life of the substance?”

i’m confused about where to start with this problem or what formula to solve it

The 'half life' is defined as the length of time it takes for your quantity to decrease by half.
Your quantity has decreased by 5.2% in 7.3 hours so you want to find out how long will it take to have decreased by 50%.
Can you determine its rate of decrease? That might be a good place to start.
Show us what you have tried in that direction and we will offer further advice if needed.

Afterthought (addendum/hint):
With the (little) information you have provided in your question I think it will be necessary to assume that the rate of decrease is constant, ie: if it has decreased by 5.2% in 7.3 hours then it will have decreased by 10.4% in 14.6 hours. (So this isn't really a "Calculus" problem. ?)

 

[Otis]

confused about where to start … or what formula

Hi ellie. Half-life is related to exponential decay. It's the amount of time required for some initial quantity to become half as much.

The exponential decay formula is:

N = N₀ • e^k•t

where k is a negative constant

t is elapsed time variable (in hours)

N₀ is the initial amount

N is the variable amount remaining at time t

They've told us that N₀ has decayed by 5.2%, so decide how to express that new amount for N (either algebraically or by picking a value for N₀, like 100).

Hint: Use the given information to find the value of k first, then substitute it in the formula and find time t required to for N to equal N₀/2.

Let us know your specific questions. Please show how far you can get.
[imath]\;[/imath]

 

[Steven G]

Actually, I don't bother to find k. Knowing what e^k•t is enough!

Here is an example that I'll work out for you.

Assume a quantity decreases by 10% in 5 hours. What is the half-life of the substance?

We know that N(t) = N₀e^kt
Want 0.9N₀ = N₀e^5k
0.9 = e^5k

We want e^kt, not e^5k. So raise both sides to the t/5 power.

Then (0.9)^t/5 = e^kt

Then N(t) = N₀(0.9)^t/5

Since we want to know when N(t) = 0.5N₀ we solve
0.5N₀= N₀(0.9)^t/5

Continue to find t.

 

[stapel]

...(So this isn't really a "Calculus" problem. ?)...

Thread moved.

“assume a quantity decreases by 5.2% in 7.3 hours. What is the half-life of the substance?”

i’m confused about where to start with this problem or what formula to solve it

Our helps and hints assume that the given topic has been covered in the original poster's textbook and in class. Replacing that lesson instruction is not reasonably feasible here. Instead, please try online lessons to obtain the necessary instruction. Then the replies above should make a lot more sense.

Eliz.