4.7 Graphical conversions: What is the relationship between the Fahrenheit and the Celsius scales?
To determine this, you first need to determine the gradient of the straight line inFigure 11.
The new vertical scale goes from 0 to 180 as the Celsius scale on the horizontal axis goes from 0 to 100. So the gradient is
and the relationship between the scales on the axes is:
value on new scale = 1.8 x value on Celsius scale. (1)
The formula tells you that there is a directly proportional relationship between the value on the Celsius scale, drawn on the horizontal axis, and the value on the new scale, drawn on the vertical axis.
You can now get the Fahrenheit scale back simply by adding 32 to the new scale, that is:
Fahrenheit scale = value on new scale + 32. (2)
Now you need something involving the Celsius scale on the right-hand side of formula (2). (Referencing formulas by numbers in brackets like (1) is part of the conventional style of the ‘written language’ of mathematics. It allows you to refer back to particular ones easily.) Notice that the words ‘value on new scale’ appear in both formula (2) and formula (1). Formula (1) deals with the Celsius scale and relates it to the new scale. The equals sign in formula (1) tells you that the words ‘value on new scale’ and ‘1.8 × value on Celsius scale’ both refer to the same number.
The words ‘value on new scale’ in formula (2) also refer to this number. Where you get different words or symbols referring to the same thing, you can replace one by another in a formula without changing the formula’s numerical value. So you can replace the words ‘value on new scale’ in formula (2) by the expression ‘1.8 × value on Celsius scale’ from formula (1) to get a new formula:
Fahrenheit scale = 1.8 Celsius scale + 32
This is the relationship you are looking for between values on the Celsius and Fahrenheit scales: the formula that represents the straight-line graph in Figure 16. You may have come across this relationship in the form of the rule ‘to convert from Celsius to Fahrenheit, multiply by 9 over 5 and add 32’. 9/5 is, of course, equal to 1.8.
You can see from Figure 16 that the graph crosses the vertical axis at 32. This number also appears in formula (3). It is called the intercept of the graph (it is where the graph intercepts the vertical axis – here it is the y-intercept). The number 1.8, also appearing in formula (3), is the gradient of the graph.
Activity 8: Cooking times
A cookery book suggests that the cooking time for chicken in an oven preheated to 180°C is calculated by allowing 20 minutes for each 0.5 kg, plus a further 20 minutes.
What is the formula relating the cooking time in minutes to the weight in kilograms?
On squared paper, draw a graph to give cooking times in minutes for weights up to 3 kg. What is the intercept and the gradient of the graph? Does this graphical relationship make sense for all weights?
A book on microwave cooking suggests that the cooking time for chicken is simply 16 minutes for each kilogram. Draw this graph on the same graph paper and use your graphs to find the cooking times for a 1.6 kg chicken in a conventional and a microwave oven.
Do not forget to scale and label the axes, and give the graph a title.
The general formula for the cooking time is
time = (time per kg x weight in kg) + extra time
Allowing 40 minutes per kilogram, plus an extra 20 minutes, gives the formula:
cooking time (minutes) = 40 x weight in kg + 20
This is the formula for a straight-line graph with a gradient of 40 and an intercept of 20. Figure 17 shows the relationship. Now check that the graph goes through the right points. When the weight is zero, the line must cross the vertical axis at 20, the value of the intercept. So one point on the graph is (0,20). Using the cooking time formula you can find another point. A 3 kg chicken will take (40×3)+20=140 minutes to cook. So (3,140) is another point on the graph.
The relationship is really a rule of thumb that has been found to work reasonably well for small whole chickens up to about 3 kg. Common sense should prevail however, and the rule may not be so useful for very small pieces, where the cooking time may be too long. By itself, the mathematics cannot make decisions for you about what is and is not reasonable. Putting a chicken of zero weight into the oven and cooking it for 20 minutes does not make a lot of sense! The same could be said if you extend the rule too far in the other direction, and try to work out the cooking times for excessively large pieces of poultry.
For microwave cooking at 16 minutes per kilogram the formula is
cooking time (minutes)16 x weight in kg
The corresponding straight line graph, which starts from the point (0, 0), is also drawn in Figure 17. In the microwave, a 3 kg chicken will take 16×3=48 minutes, so a second point on this new graph is (3,48).
The cooking times for a 1.6 kg chicken are 84 minutes (or 1 hour 24 minutes) in a conventional oven, and 26 minutes in the microwave.