Solve each problem. Nuclear Bomb DetonationSuppose the effects...

Question

Solve each problem. Nuclear Bomb DetonationSuppose the effects of detonating a nuclear bomb will be felt over a distance from the point of detonation that is directly proportional to the cube root of the yield of the bomb. Suppose a 100-kiloton bomb has certain effects to a radius of 3 km from the point of detonation. Find, to the nearest tenth, the dis-tance over which the effects would be felt for a 1500-kiloton bomb.

Accepted Answer

Hey, everyone in this problem, roses are kept in a bucket and the fragrance can be smelled over a distance from the point where the roses are kept. And that distance is directly proportional to the square root of the mass of the roses. So let's start there. We're gonna come back and figure out what this question is asking us for. But what we're told is that the distance that they can be smelt by which we're gonna call D is proportional to the square root of the mass. So it's proportional to the square root of that. Now, getting back to this problem, we're told if the fragrance of four kg of roses can be smelled over a distance of 150 m from the point where the roses are kept, find the distance over which a fragrance can be smelled for kg of roses. Now, we're given four answer choices. Option A 75 m, option B 23.71 m. Option C 237.17 m and option D 750 m. Now, in the problem, we're told that four kg of roses gives us a smell that we can smell over a distance of 100 and 50 m and we're gonna call this 500.1. OK? So the distance D one is equal to 150 m or a mass M one of four kg. Now, what we want to find is the distance D two for a mass and two of 10 kg of roses. Now, we have this proportionality where the distance is proportional to the square root of M. OK. Recall that we can relate this proportionality to an equation. OK. We can convert this into a, an equation if we introduce a proportionality constant. So what we're gonna do is replace that proportional sign by an equal sign. And then we have to multiply by that constant. So we have ad is equal to our constant K multiplied by the square root of M. Now we're gonna consider this time 0.1 first. OK? Because if we want to calculate D two right now, we substituted the mass M, but we don't know what K is. OK. So let's use this first set of information we're given where D one is equal to 100 and 50 m and M one is equal to four kg to find K. Then we can use that K value in the same equation to find D two. So substituting in our information, we have that 150 m is equal to K multiplied by the square root of four kg. So again, distance is equal to K multiplied by the square root of M. This tells us that 150 m is equal to K multiplied by two multiplied by the square root of kilogram. OK. We're gonna leave our units here just to make sure that everything checks out. We can divide both sides by two square root of kilogram to isolate for K. And we get that our proportionality constant K is gonna be equal to 75 m per square root of kilogram. So we, we've used that first set of information to figure out what this proportionality constant K is. And now we can write that our distance D is gonna be equal to 75 meters per square root kilogram multiplied by the square root of the mass M. OK? Let's not mistake the meters and the mass F now we're gonna use our second set of information, OK? To substitute in to D to find D two. OK. So what we wanna find is D two, this is gonna be equal to 75 meters divided by the square root of kilogram multiplied by the square root of our mass. In this case is 10 kg. And what you'll notice is that square root of kilogram unit divides out, we're left with just a unit of mass, which is what we want for distance. We have 75 multiplied by the square root of 10, which gives us 237. m. OK. And so the distance over which the fragrance of 10 kg of roses can be smelt is 237.17 m which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Direct Proportionality

Cube Root Function

Solving for a Constant of Proportionality

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