Aquarium Problem Extension The aquarium in the video was given as 15 feet long, 4 feet deep, and 6 feet high. The video mentioned that 4000 gallons of water weighs 32,000 pounds. It was stated that the height of the aquarium determines the thickness of the glass needed to ensure that the glass is strong enough to support the weight of the water. Thicker glass costs more, therefore, the taller the aquarium, the more expensive it will be to produce. How many gallons of water will the aquarium in the video hold? How much will the water weigh? In order to answer this, there are at least two missing facts that are necessary for us to find out. What are they? In the space below and using more paper if necessary, make a sketch of the aquarium write down what you already know and write down what you need to find out Solve the problem, making sure that your process is easily to follow, that you include units, and that your answer is well marked. 4 ft 6 ft 15 ft When you see this icon, it indicates a step towards the solution When you see this icon, it indicates that we have to stop, think critically, and make a decision that will get us closer to the solution. When you see this icon, it indicates that we are taking a slight detour to extend our answer, making it richer or making a connection that will help us in future problems. When you see this icon, it indicates that we are taking a slight detour to explain a mathematical concept that will help you with this problem and in future problems. Given: We know that the volume of the aquarium is (4ft)(6ft)(15ft) = 360 ft3 We know that 4000 gallons of water weighs 32000 pounds. What do we want to know? We want to find out how many pounds the water in the aquarium weighs. Below are the steps that I went through to solve this problem. This is not the only way to solve the problem. There may be other ways that are more efficient or more elegant. In true problem solving, you don’t always have the information that you need. You have to figure out what you do have, what you still need, figure out a way to find what you need, and then solve the problem. It’s a journey! Communication is an important part of the journey, so you can share it with others. We first need to find out how many gallons are in a cubic feet. Then we can answer the question of how many pounds the water in the aquarium weights. Our volume is in cubic feet, not gallons. Our weight is based on number of gallons, not cubic feet. Where could we find that out? The most efficient way is to look it up on the internet, but once you get there, you still have to think! I Googled “Conversion between cubic feet and gallons”. Many sites were listed. Which is the best one to go to? I looked at the first site listed and the text said “How many cubic feet in 1 gallon? The answer is 0.13368055556…” Although the answer did not have a unit (not good!), it must be cubic feet. I don’t want cubic feet, I want gallons. I could play with this fact and figure this out*(see note after the solution)….OR I could look for another site that would help me. The second site was entitled “Cubic Feet to Gallons Conversion.” This is the site that wanted: http://www.metric-conversions.org/volume/cubic-feet-to-gallons.htm When I got to the site, I was surprised to find out that there are 3 types of gallons! (U.S liquid gallons, U.S. dry gallons, and UK gallons) Which do I choose? I chose the U.S. liquid gallon. In this site, I was asked to enter the number of cubic feet. The conversion formula was automatically applied to my input of 360 cubic feet. The output was 2692.987 gallons. If all I wanted to do was answer this problem, I’ve got the answer I need from the site. For future reference, since I’m studying a lot about aquariums, it would be nice to have a conversion factor at my fingertips so that I would not have to look online each time I have a new aquarium. It would be nice to know how many gallons are in 1 cubic foot. I enter 1 cubic foot as the input and the output is 7.48051 gallons. Do I want to use so many decimal points in my conversion? It depends upon my goal for the answer…how my answer will be used. Once I start rounding (using fewer decimal points), my final answer will lose accuracy. Sometimes less accuracy is okay and sometimes accuracy is very important. 360 ft3 7.48501gal = 2694.6036 gal 1 1 ft3 360 ft3 7.49 gal 1 1 ft3 = 2696.4 gal 360 ft3 7.5 gal 1 1 ft3 = 2700 gal (If you are confused by the “canceling”, see the Detour “Multiplying by One Strategically”) Which number do we use? For our purposes, we will use the 2700 gallons, since we want to err on the side of caution…use the higher number so we don’t underestimate the weight. Now that we know that we have 2700 gallons in our aquarium, we can find the final answer to our question: How many pounds does the water in our aquarium weigh? Going back to our givens, we know that 4000 gallons of water weighs 32000 pounds. Above we converted between cubic feet and gallons. Now we need to convert between gallons and pounds. We can use the same process of canceling out units by using our conversion Do we want “gallons per pound” or “pounds per gallon”? The best unit rate to find here would be “pounds per gallon”, since we will are given the number of gallons and we want to find the weight. Let’s start by setting up the rate that will give us “pounds per gallon”, which means that pounds is in the numerator and gallon is in the denominator. Use the relationship given above (4000 gallons = 32000 pounds) to form “1”, then turn it into a unit rate. (See Detour “Unit Rates” for more explanation.) 32000 pounds 32 pounds = 4000 gallons 4 gallons 8 pounds pounds = 8 1gallon gallon So, we are almost there! 2700 gallons ∙ 8 pounds = 21600 pounds gallon So the water in the tank weighs 21600 pounds! * Unit Rates Earlier in the problem, a website told us that there are 0.13368055556 cubic feet in one gallon. I mentioned that we could find out how many gallons there are in 1 cubic foot from this fact. How do we do that? One way to think about this is as a rate. When saying a rate, we often use the word “per”. When we see the word “per”, we can translate that mathematically into “divide” Familiar rates are miles per hour, miles per gallon, feet per second, etc. Rates can be expressed as fractions, which is a form of division. The first unit in the rate goes in the numerator and the second unit in the rate goes in the denominator 35 miles hour 29 miles gallon 2 feet sec ond In each case mentioned above: the number (35, 29, 2) describes the unit in the numerator (miles, miles, feet) the number describing the unit in the denominator (hours, gallons, seconds) is understood to be 1. 35 miles 1 hour 29 miles 1 gallon 2 feet 1 sec ond These rates are called “unit rates”. We write them without the “1” 35 miles feet miles or 29 or 2 hour sec ond gallon Vocabulary Alert!! The word “unit” used in the term “unit rate” is referring to the number “one”. This can be confusing. Above, we used the word “unit” in two different contexts. We said that the units in the numerator were “miles, miles, and feet” and the units in the denominator were “hours, gallons, and seconds”. Here we were using the word “unit” to refer to “labels” or “units of measure”. Another definition of “unit” is “one”. Think of the base ten number system. In the number 432.5, the number 2 can be said to the in the “ones” place or in the “units” place. Unit rates always have an “understood 1” that describes the unit in the denominator. These unit rates are so common that abbreviations are used (mph, mpg, fps) So 35 mph means 35 miles in 1 hour 29 mpg means 29 miles for 1 gallon Now, back to the fact that there are 0.13368055556 cubic feet in one gallon. As a unit rate, this can be written 0.13368055556 ft 3 1 gallon or 0.1336805556 Now, let’s go back to the unit rate ft 3 gallon 2 feet 1 sec ond If we travel 2 feet in 1 second, how many seconds does it take to go 1 foot? It takes ½ seconds or .5 seconds to go 1 foot. This is an easy one to figure out in your head, but what we did mathematically was to turn the rate above upside down. 1sec ond 2 feet Now, take the numbers out front and keep the units as a fraction or .5 1 sec onds 2 foot sec onds foot When we switched the units in the numerator and denominator, their associated numbers had to be switch as well. In effect, we found the reciprocal. Back to 0.13368055556 ft 3 1 gallon To find how many gallons per ft3, interpret “per” to mean “divide” and rewrite the above as 1gallon 0.13368055556 ft 3 1 gallon 0.13368055556 ft 3 gallon simplify by performing the division 7.48051948 ft 3 This matches the conversion we were given from the website. take your numbers out front “Let your Units do the work for you!” Cancelling Units as a Way to Convert between Measurements Above, we used the word “unit” in two ways: as a label as the value “one” When we use units to convert between measurements, we use both of these definitions for “unit”. We place the units strategically so that they will “cancel” (i.e. form “1”) We multiply by “1” in disguised forms. Units are words, but they can be “cancelled” just like numbers can be “cancelled”. We can use this fact to determine what form of “1” that we want to multiply by in order to convert from one unit to another. By “cancel”, we mean that, in a fraction, if we have a number as a factor in the numerator and the same number as a factor in the denominator that we can use the commutative and associative properties of multiplication to move them around and express them as “1”. Example with numbers: 3 4 3 4 4 3 4 4 can be written as , which is or 1 = 53 35 5 3 5 5 Rather than go through these steps, we can cancel them out early… 3 4 4 = 53 5 We can also cancel units and doing this helps us to solve many real world problems. If we cancel units strategically, the units will tell us whether we need to multiply or divide! We don’t have to think as much! Convert Between Measurements by Multiplying by One Strategically (i.e. set up your units so that you can cancel them) Notice how we are converting from ft3 to gallons below. We are “multiplying by one”…making sure that our units “cancel”. By “multiplying by one”, I mean that the numerator and the denominator in the 3 fraction are equivalent. We know that a fraction in the form of 1 3 In the same way, even if we are looking at things that seem different, such as gallons and ft3, we know that 7.48501 gallons = 1 ft3 Therefore, because the numerator and denominator are equal, 7.48501gallons =1 1 ft 3 So we can multiply by this form of 1 any way that will help us to get the answer we are looking for. We used the above form of “1” when we solved our problem. Here is another example that doesn’t use rates, but only uses units. the fraction Let’s say we want to convert 48 inches to yards. We start with inches and strategically multiply by 1 in order to end up with yards. 48 inches 1 foot 1yard 12inches 3 feet If a number is on top, we multiply. If a number is on the bottom, we divide. We can ignore the 1’s since multiplying by 1 does not change the value. 48 ÷12 = 4 4 ÷ 3 = 1.333 So 48 inches = 1.333 yards (We could have also used the fact that 1 yard = 36 inches and used only one conversion fraction.) When we use rates, which have a unit on top and on bottom, the process is the same. Change 60 miles per hour to feet per second. 60miles 1hour 1 min 5280 feet hour 60 min 60 sec 1mile Once we do all of the multiplication and division and cancel units, we have our answer. We didn’t have to think, “Do I multiply? Do I divide?” The placement of the units to form your “ones” so that you can cancel units automatically have you do the correct operation. 60 5280 feet feet = 88 60 60 sec ond sec ond