Triginometric preliminaries

Introduction

A farmer observes that the productivity of her best milking cow, Daisy, varies throughout the calendar year.

She believes that two major factors contribute to the variation in the cow’s daily milk production.

Task 1

One factor is the stage of the cow’s pregnancy and subsequently the age of her calf.

The contribution to milk production, , in litres due to this factor is given by:

where t is the time in months since 1st January.

(a) What is the maximum milk production due to this factor?

(b) What is the period of this cycle in milk production?

The second factor is the availability of good food.

The contribution to milk production, , in litres due to this factor is given by:

where t is the time in months since 1st January.

(c) What is the maximum milk production due to this factor?

(d) What is the period of this cycle in milk production?

(e) On the same set of coordinate axes accurately draw the graphs of and for .

(f) Write an expression for , the cow’s actual daily milk production (in litres), given that it is the sum of these two factors.

(g) On a different set of coordinate axes accurately draw the graph of for .

The farmer examined this graph carefully and concluded that Daisy’s daily milk production can be modelled by the function:

.

(h) Use the graph from part (g) to find an equation for in the form:

(i) What are Daisy’s daily maximum and minimum productions and when do they occur?

(j) If the premium daily milk production the farmer expects from his cows is 20 litres, for what percentage of the year is Daisy’s production above this level?

Task 2

The farmer had already observed that daisy’s milk production was periodic and could be modelled by a function of the form:

.

(a) Using the addition formula for , write an alternative expression for .

(b) By equating this expression for to the expression found in part (a) state to 3 decimal places the values of:

;

.

(c) Using the values found in step (b), find the value of , to 3 decimal places.

(d) Using the identity , show that .

(e) Using your graphics calculator draw an accurate graph of for and compare this result to the expression for found in Task 1.

Task 3

(a) State the limitations and assumptions inherent in this model.

Comment on specific features of all three functions: .

(b) Consider 2 or 3 other factors that may affect milk production. Use some real world

predictions, explain the link of these with parameters of another function such as AsinB(x-C)+D to be added to your model. Use technology to display/develop this model to predict the resultant milk production. Discuss aspect of the changed model similar to part (a), in particular why you believe you have the improved/reduced accuracy.

(c) Determine if the farmer could improve the maximum milk production for Daisy by altering the timing of the birth of her calf.

Discuss in depth your reasoning and the outcome.

(d) Discuss whether increasing the maximum daily production would be likely to increase the total yearly production of milk.

Explain your reasoning.

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