If the density of a certain spherical atomic nucleus is 1.0 x 10 14 g/cm 3 and its mass is 3.5 x 10 -23 g, what is the radius in angstroms? (Å= 10 −10 m)

So here our practice question states, if the density of a certain spherical atomic nucleus is 1.0 times 10 to the 14 grams per centimeters cubed and its mass is 3.5 times 10 to the negative 23 grams, what is the radius in angstroms? Now, one angstrom, which is symbolized by this a with a circle on top equals 10 to the negative 10 meters. So So that's gonna come in handy when we have to do any type of metric prefix conversion associated with angstroms. Now here they're telling us it's a sphere and we're talking about its radius. Now remember the volume equation for sphere is volume equals 4 over 3 times pi times radius cubed. That means we need to isolate radius. The only way to isolate that radius is to figure out what my volume will be. Now, from the information given to us, presented to us, this is only 1 unit by itself, so that has to be our given amount. And then here we have our density, which we've said numerous times at this point, that represents a conversion factor. So let's use those. So we have 3.5 times 10 to the negative 23 grams. We're bringing our conversion factor here. So our conversion factor here, we're going to have grams canceling out, so 1.0 times 10 to the 14 grams has to go on the bottom, and then 1 centimeters cubed has to go on top. Grams here would cancel out, and we'd have centimeters cubed for our volume. So that comes out to 3.5 times 10 to the negative 37 centimeters cubed. So we're going to plug it into the formula. So 3.5 times 10 to the negative 37 centimeters cubed, and that equals 4 over 3 times pi times radius cubed. Alright. So we're going to first multiply both sides by 3 to cancel out this 3. So when I multiply this by 3, I'm going to get 1.05 times 10 to the negative 36 centimeters cubed, and that's equal to 4 times pi times radius cubed. Next, divide both sides by 4. Remember, we're trying to isolate radius here. So when I do that, I'm going to get Oh. We're going to divide it by 4 and pi at the same time. Just kill 2 birds with 1 stone. So that would be radius cubed equals 8.3 556 times 10 to the negative 38 centimeters cubed. But we wanna get rid of centimeters cubed, so we're gonna take the cube root of both sides. So for a lot of you on your calculators, you'll see this symbol, an x with a square root sign. So in your calculator you would hit the button 3, then you would hit this button, in parenthesis, you would put this value of 8.356 times 10 to the negative 38, close parenthesis, and that would give you as your radius 4.37 times 10 to the negative 13 centimeters, but realize here here that we don't want the radius in centimeters, we want it in angstroms. So we have to do a metric prefix conversion. So we're gonna have here 34.37 times 10 to the negative 13 centimeters. Centimeters go on the bottom, our base of meters goes on top. One centi is 10 to the negative 2. So now we have meters, and then let's use this conversion factor here. Meters goes on the bottom, angstroms go on top. One angstrom is 10 to the negative 10 meters. So if you do that correctly, you should get as your answer 4.3 7 times 10 to the negative 5 angstroms. We're going to say here that this number coefficient has 2 sig figs, and 3.5 has 2 sig figs. So the best answer here would be 4.4 times 10 to the negative 5 angstroms. So remember, when dealing with geometric objects, we commonly run into spheres and cubes and cylinders. Their formulas oftentimes do not need to be memorized. They're usually given to you within the question or on a formula sheet. But to be absolutely sure, always consult with your professor. Now remember, even though it's density, we do take components from dimensional analysis to help guide us to our end amount. So keep in mind what we've learned thus far about conversion factors, given amounts, end amounts, and how it all applies to dimensional analysis and, of course, density.