More or less?
Are these estimates of physical quantities accurate?
Age
14 to 16
Challenge level
Problem
A set of estimates of various physical quantities is shown below. For each estimate, consider the following three questions:
- Do you think that the estimation will be an over- or under-estimate, or will it be essentially exactly correct? Or will it be impossible to say without more information? Be as clear as you can with your reasoning.
- Why do you think that the solver made the estimate in this way? What assumptions were made? Can you reproduce the calculation?
- How close do you think the estimations would be to the real values?
Note that you might need to use standard scientific data not provided in the questions for some of the numbers used and you might need to use a calculator. You will certainly need to decide which formulae to use to relate the quantities in the question.
- I wish to estimate the volume of an apple. It weighs 76.2g. I therefore estimate its volume to be 76.2cm$^3$.
- I have a set of ball bearings of volume 1cm$^3$. A large crate is filled to the brim with ball bearings and closed with a lid. The box contains 850 ball bearings, so I estimate that the box has a volume of 850cm$^3$.
- An oak tree measures 106cm around the base, corresponding to a cross section of 0.0894m$^2$. I have measured the height to be 7.3m. I estimate the volume to be 0.65m$^3$. How would your answer differ if the question related to a fir tree (with the same numbers)?
- In a wood of area 27000m$^2$ I find 17 earthworms in a volume of soil of 1m$^2$ in area and 20 cm deep. I therefore estimate that there are 459000 earthworms in the wood.
- A cheetah can run at 100km h$^{-1}$, so in a 3 minute chase of prey I estimate that it might be able to run 5km.
- A 10 kg sample of sea water taken from a point far from the shore contains 350g of salts. Therefore, to obtain 100kg of salts I would need to take a sample of 3000kg of sea water.
- A cylindrical container barrel has a capacity of 700l. To obtain the 100kg of salts required in question 6, I would only need to fill 4 barrels.
- A bacterial culture initially contained 7 cells. After 35 minutes it contained 14 cells and after 1 hour 10 minutes it contained 28 cells. I estimate that after a further 12 hours there will be around 43.6 million cells.
- I catch 100 adult fish from a lake and mark them all with a tag. One week later I catch 100 adult fish from the lake and find that 12 of them are marked. I estimate that there are 800 adult fish in the lake.
- Light of a certain wavelength is shone through a solution of plant cells. 90% of this light is absorbed by the solution. If this solution was diluted 1 part solution to 9 parts water then I estimate that 10% of the same light would be absorbed. A second solution absorbs 100% of this light. I estimate that a 10% dilution would absorb 10% of the light.
NOTES AND BACKGROUND
Making sensible approximations is a key skill used by anyone involved in the application of mathematics. A good approximation can help to catch errors and reduce the time taken to complete subsequent, more refined calculations. A skilled applied mathematician will always be aware of the level of approximation being applied to a problem, will develop a feel for the range of sensible approximations and will know the mathematics required to justify the approximations.
Making sensible approximations is a key skill used by anyone involved in the application of mathematics. A good approximation can help to catch errors and reduce the time taken to complete subsequent, more refined calculations. A skilled applied mathematician will always be aware of the level of approximation being applied to a problem, will develop a feel for the range of sensible approximations and will know the mathematics required to justify the approximations.
Getting Started
In each question there will be some way of relating the quantities referred to. You will need to think what equations from science to use.
Once you have the equations, what quantities don't you know? Could you substitute sensible values for one of these?
Don't forget to use physical intuition. For example, an apple floats in water. What does that tell you about its density?
Student Solutions
Patrick from Woodbridge School gave a really good solution to this problem. There is much to consider in this problem, so we have added a few editorial notes of interest to Patrick's solution.
1. This is a reasonable assumption, given that an apple is mostly made up of water with a density of $1$g cm$^{-3}$. [This is an over-estimate: since apples float, their density will be a bit less than that of water.]
2. This is clearly an under-estimate since spheres do not tessellate at all well (there is empty space between each sphere). It will be out by some considerable amount. It would be much easier to test the volume by filling it with water and measuring that. [Roughly speaking you can fill up to about $74\%$ of space using spheres. So the volume will be about $850/0.74 =1150$ cm$^3$.]
3. $0.0894 \times 7.3 = 0.65$ gives the rough volume of the trunk (this will be inaccurate as the trunk tapers, so it will be an overestimate, but it will not be a very big difference). However, the branches make up a lot of the tree's volume so it will be considerably under the true value. A fir tree has fewer branches, so it will still be an underestimate, but not so far out. [it is a good point that the branches are an important factor in the weight of the tree, not just the shape of the trunk.]
4. $27000\times 17=459000$ so this would be a reasonable assumption for the first $20$cm of soil. However, this fails to take into account the possibility that there is much more soil underneath the tested area, so the estimate is probably a very long way under the true value. [This assumes that soil is uniform. Some areas might be more or less fertile than this. Also we would need to know how far down the worms are likely to live.]
5. The cheetah's top speed is 100kmh$^{-1}$, but it cannot sustain this for three minutes - it is only a peak measurement. Therefore this distance is an overestimate (but not by very much). [$3$ mins at $100$kph$^{-1} = 100\times 3/60 =5$. Can a cheetah maintain top speed for 3 mins?]
6. This is a perfectly reasonable estimate. It is likely to be almost exactly right.[Assume salt concentration and water density is uniform: this is very likely. If $10$kg of water gives $0.35$kg of salt then $10\times 100/0.35 =2857$ kg of water gives $100$ kg of salt.]
7. We need more information to solve this question - I have no idea what the density of the salts is. [Four barrels will contain up to $2800$l of fluid. $3000$kg of sea water would fit providing that the density is greater than $3000/2800 = 1.07$ kg/l. The actual figure is about $1027$ kg m$^{-3}$, so the required amount of salt water will easily fit.]
8. Apparently every cell divides in two every $35$ minutes. $12$ hrs = $720$ minutes; $7200/35 = 20.57$. So the cells increase by a factor of $2^{20.57}$, which gives $43.6$ million. [This figure is probably an upper bound, as it does not take into account issues concerning possible death rate, increasing scarcity of food or other changing environmental factors.]
9. The reasoning is that $100/12 = 8.33$, so the estimate for fish will be $100\times 8.33 = 833$. I would therefore say that the estimate is a little under the correct value. [This proportional reasoning is a good first estimate. More advanced statistical considerations would give rise to a likely range of values. Of course, the estimate does not take into account environmental factors such as fish being eaten, moving away from the test area, tagged fish being possibly more likely to be captured etc.]
10. I think this sounds like it should be correct. [This is likely to be true for the first part, but not necessarily true for solutions which absorb $100\%$ of the light. Imagine a jug of black ink. Diluting this ink might still yield a completely opaque jug of liquid until quite a large volume of water has been added.]
Teachers' Resources
Why do this problem ?
This problem gives practice with the use of estimating numbers and making choices about the information to use in a question. These are crucial mathematical skills in the sciences. These interesting questions will allow students to practise these skills whilst developing awareness of orders of magnitude in scientific contexts. As with any problems involving approximation, they offer opportunity for classroom discussion and justification.Possible approach
There are several parts to this question, arranged in approximate order of difficulty. The individual pieces could be used as starters or filler activities for students who finish classwork early. Enthusiastic students might work through them in their own time. Since there is no absolutely 'correct' answer to many of these questions, they might productively be used for discussion: students
create their own answers and then explain them to the rest of the class. Does the class agree? Disagree? Is there an obvious best 'collective' answer?
The questions are arranged for printing out into cards in this Word 2003 document .
Key questions
- What assumptions will you need to make in this question?
- What formulae will you need to use?
- How accurate do you think the answer is?
- What 'order of magnitude' checks could you make to test that your answer is sensible?