Question:Question 2

In this question, where necessary, you should round answers appropriately.

(a) A researcher claims that the average six-year-old child has a vocabulary of around five thousand words and subsequently acquires three thousand words per year until they are 12 years old.

The size of the average child's vocabulary over this period can be modelled by the equation

v = 3000(t ? 6) + 5000 (6 ? t ? 12), where v is the size of vocabulary in number of words, and t is age in years.

(i) Find the vocabulary size of a nine-year-old according to this model.

(ii) Explain the inequality (6 ? t ? 12) that follows the equation.

(iii) Using algebra, calculate the age at which the vocabulary size is twenty thousand words.

(iv) Write down the gradient of the straight line represented by the equation v = 3000(t ? 6) + 5000. What does this measure in the practical situation being modelled?

(v) Explain why the vertical v-intercept is not 5000.

(vi) Either using Graphplotter or by hand, sketch the graph of

v = 3000(t ? 6) + 5000, putting v on the vertical axis and covering the time interval 6 ? t ? 12.

(b) A mother with an interest in language development records the number of words that her child can say, up to the age of 36 months. She claims that the size of her child's vocabulary can be modelled by an exponential model with equation v = 0.27 (1.26)t , where v is the size of vocabulary and t is age in months.

(i) Assuming that the model applies for any given child up to the age of 36 months, calculate the size of vocabulary for a child aged 18 months and a child aged 30 months.

(ii) Write down the scale factor, and use this to find the percentage increase in vocabulary each month.

(iii) Use the method shown in Unit 13, Subsection 5.2 to find the age at which a child will acquire a vocabulary of five hundred words.

(Example of Unit 13 Subsection 5.2)

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