### Elastic Maths

How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.

### Logic, Truth Tables and Switching Circuits Challenge

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record your findings.

### Truth Tables and Electronic Circuits

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

# Scientific Measurement

##### Stage: 4 Challenge Level:

In the following tables we provide the numerical answers to the first four questions associated with each picture. The remaining last question is randomly generated by the program, so below we give an outline of the strategy one needs to follow to tackle it.

In general, it is important to take more than one measurements of each quantity, in order to improve accuracy. Questions 1 and 2 in each problem are about making a few measurements on the picture, and question 3 involves making a simple computation using these measurements. Questions 4 and 5 are not related to the image, and only require an accurate measurement of the scale bar and then some calculations about the scale of the picture.

As an example, we will solve the fifth set of questions, which are about the Pollen. Initially we measure the scale bar to find it has a length of 2.1 units.

Question 1

We find that the diameter of the particle A is 3.2 units, so we take that the radius we require is 1.6 units. To convert this into meters, we compare it with the scale bar, and so we see that the radius is

$$r_A = \frac{1.6}{2.1} \times 0.00006 \approx 4.57 \times 10^{-5}$$

This gives us a percent error of 1.7 which is an excellent estimate. Now, with some trial and error, we find that the most accurate value we can get is $4.5 \times 10^{-5}$.

Question 2

The diameter of particle B is 0.7 units, and since this is exactly 1 third of the scale bar, we immediately conclude that particle B has a diameter of 0.00002 and hence a radius of 0.00001 meters. Trying this value, we get a percent error of 0, so our approximation is exact.

Question 3

First of all, we compare the volume of A to the volume of B. In particular, we get

$$\frac{V_A}{V_B} = {r_A^3}{r_B^3} \approx 91.13$$

So, if we assume a packing efficiency of 74%, we expect that approximately $91.13 \times 0.74 \approx 67$ B particles will fit inside a particle A. Our approximation is not absolutely correct, but again a few trials should yield the correct answer, which is 68 particles.

Now, we present the tables with the exact numerical answers to all the questions. At the end we have the strategy for solving the fifth question in each set.

Table 1: Cholera

 1 0.2 2 0.1 3 24 4 17000

Table 2: Polio

 1 0.000000009 2 110 $\times$ 10$^6$ 3 1 $\times$ 10$^{17}$ 4 840000

Table 3: Tobacco mosaic virus

 1 2.9e-07 2 34000 3 1.09e+16 4 100000

Table 4: Mammalian mitochondria

 1 5.5e-07 2 1800 3 0.25 4 71000

Table 5: Pollen

 1 4.5 $\times$ 10$^{-5}$ 2 1 $\times$ 10$^{-5}$ 3 68 4 350

Table 6: Escherichia Coli

 1 1.6 $\times$ 10$^{-6}$ 2 3 $\times$ 10$^{-19}$ 3 25 4 16000

Table 7: Human chromosomes

 1 2 $\times$ 10$^{-6}$ 2 9 $\times$ 10$^{-6}$ 3 4.5 $\times$ 10$^7$ 4 1600

Table 8: Diatom

 1 2.1 $\times$ 10$^{-5}$ 2 9 $\times$ 10$^{9}$ 3 3000 4 800

Table 9: Asterionella formosa

 1 1.6 $\times$ 10$^{-3}$ 2 0.09 3 220 4 20

Table 10: Drosophila melanogaster

 1 2.5 2 0.06 3 1.1 $\times$ 10$^{-3}$ 4 40

Table 11: Plagiomnium affine chloroplasts

 1 5.7 $\times$ 10$^{-5}$ 2 1.5 $\times$ 10$^{-9}$ 3 410000 4 700

Table 12: Whale

 1 5.7 2 24 3 67 4 5 $\times$ 10$^{-3}$

Outline of solution for Question 5:

The ruler measures cm. So, the scale bar represents a length of

$$L = \frac{\textrm{length in cm}\times 10^{-2}}{\textrm{Magnification}} \textrm{meters}$$

These are the images. They are here for convenience so that they can be edited

Cholera
Whale
Polio