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- Solutions 12

A copper refinery produces a copper ingot weighing 150 lb. If the copper is drawn into wire whose diameter is 7.50 mm, how many feet of copper can be obtained from the ingot? The density of copper is 8.94 g/cm^{3}. (Assume that the wire is a cylinder whose volume V = πr ^{2}h , where r is its radius and h is its height or length.)

Okay guys, so in this question we have a copper ingot that weights 150 pounds. Then they tell us that the copper is drawn into a wire who's diameter is 7.50 millimeters, they want to know how many feet of copper can be from the ingot. So basically we are looking for the length in feet, they tell us that the density is going to be 8.94 grams per cubic centimeters and they tell us the formula for volume which is pi R squared height or H, H in this case can be height or length so we're looking for the H, because we're looking for the length in feet. So the first step here would be to figure out the mass because if we have the mass then we can find the volume using the density, you could find the volume then we could plug everything that we know into our equation for volume to figure out the length. So I am going to remove myself from the shot because we're going to need more space to work with. Like I said the first step is to figure out the mass , so we have 150 pounds and we need to figure out in grams. We know that 1 pound equals 453.6 grams, we put pounds on the bottom because we want to be able to cancel it out, then multiply the two and you guys should get 6.80 times 10 to the 4 grams. To find the volume we use density, so density is going to be mass over volume, if we want to find the volume. We take our mass and we divide by the density, so lets do that now. So volume equals the mass which we just found which is 6.80 times 10 to the 4 grams divide it by the density which is 8.94 grams per cubic centimeters. So not only does the formula tell us what to do but also the units hint us as what we should do. So we want to cancel out the units of grams, because we want the units of volume which is cubic centimeters so that's when we place the grams on the bottom, so we're able to cancel it out. So if you did by chance forget that volume equals mass divided by density you can figure out the answer just by looking at the units. So when you guys do the math you should get 7.61 times 10 to the 3 cubic centimeters. Next step, is to find the radius, the radius we find from the diameter because its going to be the diameter divided by 2, so we have 7.50 millimeters and we divide them by 2. So the radius is going to be 3.75 millimeters, well we don't want millimeters because their volume is in centimeters, so we want to convert millimeters to centimeters. We can't convert directly from millimeters to centimeters so we are going to have to go through the base unit meter, so we first convert millimeters to meters. So we have 1 here which is in front of the prefix as milli is 10 to the negative 3 then we're going to convert from meters to centimeters, 1 goes in front of the prefix and centi is 10 to the negative 2.

So we replace millimeters on the bottom because we're able to cancel out and we're also able to cancel out meters with meters. So we're left with the unit of centimeters, so when you multiply everything on the top and divide by everything on the bottom your radius is 0.375 centimeters. So now we can plug everything into our equation for volume which is volume equals pi R squared height. So our volume is 7.61 times 10 to the 3 cubic centimeters, pi is 3.14, radius we just got is 0.375 centimeters squared and we are looking for the length. Remember, when you are squaring here, you are squaring the volume as well as the unit so on this side you have 7.61 times 10 to the 3 cubic centimeters and when you do the upper ratio here you guys should get 0.442 centimeter squared times height or length. So divide both sides by 0.442 centimeters squared, so your length should be 1.72 times 10 to the 4 centimeters. Here we have centimeters cubed and we have centimeters squared, so now we're only going to have 1 centimeters left. The question is asking us to find the length in feet not centimeters, so the next step or the final step is to convert centimeters to feet so we have 1.72 times 10 to the 4 centimeters. First, we convert to meters using our prefix, right centi goes on the bottom to be able to cancel out the units, 1 goes wherever there's a prefix and the unit centi means 10 to the negative 2 from meters. We're going to go kilometers using our prefix once again meters goes on the bottom to cancel out with meters, 1 goes wherever there's a prefix and kilo is 10 to the 3. Finally, we know that we have 5280 feet for every 1.6093 kilometers, can we place kilometers on the bottom to cancel out with the units. So now we're left with the units of feet which is what we're looking for multiply everything on the top and divide by everything on the bottom. You guys should get 564 feet as your final answer, alright see you guys soon.