Estimating Using Powers of Ten

On Thursday, my teacher decided on teaching us how to estimate using powers of ten.  On an unrelated topic, before we began, our class switched seats, so I'm with a new group.  I like this group, but I also liked my old one.  We had some fun times together, but I think this group will also have its fun moments.

To get back on topic, my teacher showed us the powers of ten first.  After all, you can't do anything unless you know the fundamentals.

Powers of ten:

10^-3= 0.001

10^-2= 0.01

10^-1= 0.1

10^0= 1

10^1= 10

10^2= 100

10^3= 1000

These powers of ten go on and on in both directions, but we decided to just stick with the simplest ones for now.  However, our class got confused when we were asked to round. What if our number is 500? Would we choose 10^2 as the estimation or 10^3?  To solve this questions, we figured out what 10^0.5 would be. That number is in between 10^0 and 10^1, so it should give us the number at which we would round up. Everything below that number we would round down.  The number came out to be 3.16.  We then found 10^1.5 which is 31.6, and 10^2.5, which is 316, and so on and so forth.  Basically, we now know that, for the number 500 for example, we would round up to 10^3 because it is more than 316.

Wacky Question 1

Our teacher then decided to use the wackiest questions to help teach us how to estimate using the powers of ten.  Our first question was, "How many basketballs could fit into the gym?"  Weird question right? However, it was actually quite fun to figure out.

My group and I estimated the length, width, and height of the gym to the powers of ten.

Gym Estimations:

Length: 10^1 meters

Width: 10^1 meters

Height: 10^1 meters

To figure this out, all we did was choose the most logical answer.  We then did the same thing with the diameter of the basketball, which we found to logically be 10^-1 meters.  However, what we really needed was the volume of the basketball and the gym.  For the gym, you would multiply the three dimensions and get 10^3 meters.  For the basketball, some confusion came about.  Do we put the ball into a cube and estimate the volume of the cube?  That would be pretty close to the volume of the basketball.  Or, do we use the actual formula for the volume of a sphere to be more precise?  My teacher didn't seem to mind either way, so we tried both.  Using the cube method, we got 10^-3 meters, but using the formula method, we got 10^-2 meters.  We didn't really discuss what to do when this happens, so we just figured out how many basketballs would fit in the gym using both.  To find the answer, you would do 10^3/10^-2 or 10^3/10^-3 and you would get either 10^5 basketballs or 10^6 basketballs.  Basically, you would divide the volume of the gym by the volume of a basketball to find out how many basketball would fit inside the gym.

Wacky Question 2

We also used the same process when my teacher asked us to figure out how many marbles would fit into the average UCS school bus.  Because I already showed you the process, I'm not going to do it again, but just for fun, the answer came out to be 10^7 marbles or 10,000,000 marbles.  Pretty big number right?  That would probably be really expensive to buy all the marbled to fit into one school bus.

Helpful Hints

• When dividing with numbers with exponents, subtract the exponents to get your answer.  For example, when doing 10^3/10^-3, you would do 3+ (-3) for your exponents, and your final answer would be 10^6.
• When rounding, if the number is greater than 3.16, 31.6, 316, and so on, then round up a power of ten, but if the number is less, round down a number of ten.
• When multiplying numbers with exponents, add the exponents together.  If you have 10^1 times 10^1 times 10^0, you would do 1+ 1+ 0, making your final answer, 10^2.
• Knowing the actual dimensions helps when estimating.

Reflection

Out of all the classes I have had so far in physics, I personally believe that this one was the most confusing.  I understood the general idea, but I don't think I have a firm grasp on it yet.  I've decided that I will practice until I get it down, just in case I need to use it again, which I probably will.