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Problem 24 Hard Difficulty

A building has become accidentally contaminated with radioactivity. The longest - lived material in the building is strontium - 90. (The atomic mass of $\underset{38}{90} \mathrm{Sr}$ is 89.907 7 u.) If the
building initially contained 5.0 kg of this substance and the safe level is less than 10.0 counts/min, how long will the building be unsafe?

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Top Physics 103 Educators
Elyse G.

Cornell University

Zachary M.

Hope College

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University of Sheffield

Jared E.

University of Winnipeg

Video Transcript

for number 24. We have strong team 90 uh, which has a massive 89.9577 years. We started with five kilograms of it. Everyone know how long until it's safe and it's gonna be safe when has an activity of 10 counts per minute? Um, in the back in appendix B, I looked up the half life on the half life with 29.1 years. We do a couple things that ahead of time here that I know. I'm gonna have to dio um so we don't get hung up with that when we do the equation. I'm giving the mass five kilograms, but I want to figure out how many nuclear that actually is. Some starting with five kilograms. And that's someone changed it. The use, first of all, So 1.66 times 10 to the negative 27th kilograms is one you. And then I know for this strong Liam that there's 89.9077 use for each nuclear. So my kilograms canceled, you canceled, and I'll be left a nuclear. So this is 3.35 times 10 to the 25th come down here. This is a weird nuclear. And now I'm going to convert my half life in two minutes. Um, I starting 29.1 years. I know that one year is 365 days. You know that one day is 24 hours on that one hour is 60 minutes. So year canceled. Day canceled our canceled, and I'm gonna be in minutes. I'm gonna put that right up underneath this, like, showed my conversion year. That that's gonna be, um, 1.5 to 9 times 10 to the seventh. And the reason I want two minutes and not down the seconds because my count here isn't 10 counts per minute. Someone tried to stay at minutes. All right, so now from my equation, um, the question I like to use for pretty much all of these is the number of nuclei is the original number of nuclear times 1/2 raised to the number of half life's you've had guys I just found My original number of nuclear is 3.35 times 10 of the 25. So 1/2 raised to the number of half lives. And I wouldn't know how long it takes to get to this activity. So I'm gonna know how much time. I'm just gonna call that X, I guess. Out of harmony, You know, how many years out of half lives with my half life? I'm just gonna keep it in years. So my answer here is going to turn out to be in years, you know, as long as either the same. It works out. So I'm gonna put 29.1 in here. So if I divide that, you know, by half lives, that'll tell me the number of half lifes I have, It's not over here. This end is the number of nuclear I'm gonna end up with. But I don't want that. Have the activity I want to end up. This is gonna be our that I want. So I need another relationship here. I know that our is equal to the decay rate times the number of nuclear. Okay. And then I know that the decay rate, though, is the half life over the natural luggage too. Uh, which is 0.693 Paralytic it. Right. So I'm gonna start here and find the decay rate. Then I'm gonna come here and find an expression for my my are really my n um, for here. I'm gonna just rearrange this, Know that, um, and would be or over the decay rate. So that's where I'm heading here. I want this to be our over Dickie, right. I'm just gonna write myself a note here. I think this is gonna be our Listen. Aargh! Let's go. We are over the decay rate. So again, I'm just gonna keep, um, that in minutes. I think my r I can keep that in minutes. Here, that is. I want 10 counts per minute. So why are will be 10. I just need this Dick Avery in minutes now. So that's why I took my half life. You're down two minutes. So my hand off life, There's 1.5 to 9 times 10 to the seventh, so my decay rate in terms of minutes will be four point 53 times 10 of the negativity. Soon I come over here, remember? This is or over my decorate my are that I want is 10 counts per minute. So my decay rate is gonna be this 4.53 Tempers tend the negative 18th. So I just need to solve this equation for X. Um, hi. Divided. And they divide both sides by this. So I get six point 59 Tim's turned the negative. 18th is equal to 1/2 raised to the X over 29.1. I'm gonna take the log of both sides, so it's gonna be the log of this. But a log of this remembering you raised her exponents, you could bring it in front of the log. So this is gonna be X over 29.1 times. Log of 1/2 call 8.5. So now, from my mouth, I'm just gonna divide, you know, the longer that divided by the walk of this times 29.1. And since I kept that in years, my ex will be in years. I got home 16. 60 and that's years

University of Virginia
Top Physics 103 Educators
Elyse G.

Cornell University

Zachary M.

Hope College

Aspen F.

University of Sheffield

Jared E.

University of Winnipeg