Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.
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Final Answer

3.99 cm3.99 \textrm{ cm}

Solution video

OpenStax College Physics for AP® Courses, Chapter 11, Problem 5 (Problems & Exercises)

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Video Transcript
This is College Physics Answers with Shaun Dychko. Knowing that this coffee cup is holding 375 grams of coffee and we assume the coffee density is the same as that of water— which is 1.00 gram per milliliter, which we convert into 1.00 gram per cubic centimeter by multiplying by 1 milliliter for every cubic centimeter— and the cup is filled to a height of 7.50 centimeters, we are going to figure out what is the inside radius of this coffee cup, which is to say the radius just of the liquid itself up until it just touches the inside rim of the coffee cup? And we are going to leave our units as grams and centimeters which is unusual... normally, we would convert into meters, kilograms and seconds these m.k.s units are standard for formulas but in this case, it is going to work together nicely as we'll see later just for convenience we will leave it as grams and centimeters. If you converted them to kilograms and meters that would be totally fine... you would have to convert this into kilograms per cubic meter as well. Okay! The volume of this cylindrical piece, if you will, of coffee is the cross-sectional area or the area of the bottom which is π times radius squared multiplied by its height—that's the volume of a cylinder. The density of coffee can be expressed as mass divided by its volume and we can multiply both sides by volume and divide both sides by the density ρ to create a second equation for the volume here, the ρ's cancel on the left and the volumes cancel on the right and we are left with volume equals mass divided by density. So now we have two equations for volume and we can equate the two together. So we have mass divided by density substituted in place of V here equals πr squaredh and we want to solve for r squared first divide both sides by πh and multiplying by the reciprocal and then you have r squared on one side and then m over πρh on the other and then take the square root of both sides and we have the radius of the cylindrical piece of water or coffee is the square root of 375 grams divided by π times 1.00 gram per cubic centimeter times 7.50 centimeters, the grams cancel... this centimeter cancels with one of the centimeters here making this centimeters squared and this is centimeters squared in the denominator of a factor within the denominator of this larger fraction... it's actually centimeters squared in the numerator overall we could multiply both sides by 1.00 centimeter squared over 1.00 centimeter squared and these centimeters squared would cancel leaving us with centimeters squared in the top which is just to say that the units work out nicely, we did not need to convert into meters, kilograms or seconds. After this calculation, we will have 3.9894 centimeters as our answer or 3.99 centimeters is the inside radius of the coffee cup.