Maths for A Level Biology - updated edition

Organisms as spheres A spherical bacterium has a diameter of 8 μm. Its area is found by the equation area = 4π r 2 and its volume by the equation volume = 4 3 π r 3 . Find the surface area : volume ratio for this bacterium. The calculations are straightforward. The diameter will provide you with the radius, and the answer is found by substituting into the equations: First the surface area: Write the equation area = 4π r 2 Find the radius radius =  8 2  = 4 μm Substitute into the equation area = 4 × π × 4 × 4 = 201 μm 2 Now ϐind the volume: Write the equation volume =  4 3 π r 3 Substitute into the equation volume =  4 3 × π × 4 × 4 × 4 = 268 μm 2 Now ϐind the ratio of area : volume ratio = 201 : 268 Simplify the ratio by dividing both sides by 268 so that the answer comes out as x  : 1 ratio = 0.75 : 1 A biological note: when ratios are simpliϐied, it is normal to divide both sides by the smaller number. When biologists study surface area : volume ratio, however, they consider unit volume, so that it is easier to compare the area different structures have in relation to their volumes. You always make the volume = 1 and so, as in the example above, you divide both sides by the numerical value for the volume. Organisms as cylinders The third shape you might encounter is a cylinder. This could represent a worm. On the right is a cylinder of radius r and length l . The circumference of the cylinder is the circumference of a circle:  C  = 2π r If you know the diameter of the cylinder, d , you can ϐind the radius because it is half of the diameter: d = 2 r To work out the area of the cylinder we can imagine it uncurled into a rectangle. The short side is the circumference of the circle and the long side is the length l . Area of rectangle = length of short side × length of long side = 2π rl Volume of a cylinder = area of circle × length = π r 2 × l Surface area : volume = 2π rl  : π r 2 l Divide both sides by π r 2 l so ratio =  2 r  : 1 Pointer Write the equation in full to start with. Pointer Simplify the ratio to give x : 1. r l l 2 π r 2 Processed numbers 39