# Does this sound about right?

A scientist makes a set of estimates of various physical quantities. Can you work out how the scientist made her estimates by reproducing the calculations? Do the answers sound about right, or has the scientist made any significant mistakes?

1. A bottle of water contains 500cm$^{3}$ of liquid. I fill a crate measuring 1m by 50cm by 50cm with bottles of water to take on a field trip. I estimate that the crate contains 500 bottles of water.

2. The number of rings a tree has on its trunk can tell you how old it is. On a tree stump I measure the distance between two adjacent rings and find that it is 0.6cm. The diameter of the stump is almost half a metre. I estimate that the tree was 42 years old when it was cut down.

3. Today I ate a 30g packet of crisps at morning break time, as I always do, so I estimate that I eat almost 11kg of crisps a year.

4. A packet of sugar weighs 1kg. My friend and I take two spoons of sugar in our coffee and we each drink 4 cups per day. One packet should just about last us for the two month field trip that we are planning.

5. My round trip to work each day is about 22 miles, but I can claim mileage from work. I estimate that I can claim for 8000 miles each year.

6. Last month the energy costs in my lab were £560. I estimate that my energy costs per year will be £7000.

7. I have a large mound of building rubble which needs disposing of. The mound is about 2 metres high in the middle and about 3 metres across. I estimate that there is about 4.7 cubic metres of rubble which needs to be removed, so I should be safe ordering a 5 cubic metre skip.

8. A very large waterfall measures 170 metres across. I measure the flow rate in the centre and it is 124 cubic metres of flow per metre per minute. I estimate that 632 metric tonnes of water flows over the waterfall per hour.

9. My vegetable plot for testing variations of plants measures 9.5 by 11 metres. I test two square metres of ground and find 53 worms in one section and 42 worms in the other. I estimate a population of 5000 worms in the vegetable plot.

**1.** **Yes** - As $ 1 \textrm{m} = 100 \textrm{cm} $ a crate with dimensions 100cm by 50cm by 50cm will have volume $ V = 250,000 \textrm{cm}^3 $. Hence, the number of water bottles required to fill this up is

$$ N = \frac{250,000}{500} = 500 $$

**2. Yes** - If the diameter of the tree is almost half a meter, its radius is almost 25 cm. Now, if the tree's radius grows by 0.6 cm in each year, in 42 years the radius should be about $ 42 \cdot 0.6 = 25.2 \textrm{cm} $, and this is about right. However, to make a safer estimate (as it is unlikely that the tree will be growing by exactly 0.6 cm each
year) a range of the form 38 - 45 years would be better.

**3. No** - While $365 \cdot 0.3 = 10.95 \textrm{kg}$, which is close to the estimate, perhaps it would be sensible to consider that the scientist only has crisps on the *working days* of the year (since she is having a pack in her morning break). Now, a person is on average working 44 weeks each year, so they have $44 \cdot 5 = 220 $ working days
(and perhaps slightly less, if we take into account bank holidays).

Therefore, the scientist is more likely to be consuming $220 \cdot 0.3 = 6.6 \textrm{kg} $ of crisps each year.

**4. No** - A teaspoon of sugar weighs about 5 grams. So, the scientist and her friend consume together 8 cups of coffee each day, with 16 teaspoons of sugar. So, each day they consume $16 \cdot 5 = 80 \textrm{g}$ of sugar. Thus, in a two - month period, they would need $ 60 \cdot 80 = 4800 \textrm{g} $ of sugar, which is much more than a packet!

**5. No** - Once again, the calculation $365 \cdot 22 = 8030 $ is correct, but 365 is not the right number to use, as she is not working every day of the year. Using our previous estimate for the number of working days in a year, we see that the scientist is actually going to claim around $220 \cdot 22 = 4840 $ miles from work.

**6. Yes** - In this case, it is highly likely that the lab will be working all year round, so the estimate $560 \cdot 12 = 6720 $ is about right.

**7. Yes** - We can model the rubble by a large cone, whose height is 2m and base radius is 1.5m. The volume of such a cone is $$ V = \frac{1}{3}\cdot \pi \cdot r^2 \cdot h = \frac{1}{3} \cdot \pi \cdot 1.5^2 \cdot 2 \approx 4.7 m^3 $$ So the estimation is correct, and a 5 cubic meter skip should be large enough.

**8. No** - $124 \cdot 170 = 21080 m^3 $ mean that each minute there are 21080 cubic meters of water flowing. Hence, over the course of an hour, there will be $21080 \cdot 60 = 1,264,800 m^3$ flowing, which is equivalent to 1,264.8 metric tonnes.

**9. Yes** - The area of the vegetable plot is $ A = 9.5 \cdot 11 = 104.5 m^2 $. Now, with the test data as described, we expect that *on average* there are $\frac {42 + 53}{2} = 47.5 $ worms per square meter.

Hence, a good estimate for the number of worms in the whole plot is $ N = 47.5 \cdot 104.5 = 4963.8 $, so the scientist is correct in this case.

**Well done to the students in Class 5C of the Brooklands Primary School, for sending us some very well-reasoned and clear answers to most of these short problems!**

### Why do this problem ?

This problem gives practice with the use of estimating numbers and deciding whether an estimated answer is reasonable. 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. If students disagree with each other, or with the answers provided, this could lead to productive discussion.The questions are available as a worksheet.

### Key questions

Do you have all the information you need to check the calculation? If not, where can you find out what you need?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?