IB Mathematics SL 2.5 Modelling with the various functions AI SL Paper 2 - Exam Style Questions - New Syllabus

(a)

Substitute \(x=10\) into the model:

\(y=-167+6264\log_{10}10\) (A1)

Since \(\log_{10}10=1\),

\(y=-167+6264=6097\).

So the model predicts that the child understands approximately \(6097\) words. A1

✅ Answer: \(6097\) words

(b)

For fluency, the child must understand at least \(5000\) words, so set \(y=5000\):

\(5000=-167+6264\log_{10}x\) (A1)

\(5167=6264\log_{10}x\)

\(\log_{10}x=\dfrac{5167}{6264}\)

\(x=10^{\frac{5167}{6264}}\)

\(x=6.681275\ldots\)

Correct to four significant figures,

\(x=6.681\). A1

✅ Answer: \(6.681\) years

(c)

On the child’s first birthday, \(x=1\).

Substitute \(x=1\) into the model:

\(y=-167+6264\log_{10}1\)

Since \(\log_{10}1=0\),

\(y=-167\). A1

This is not realistic because a child cannot understand a negative number of words. R1

✅ Answer: The model is unrealistic at \(x=1\), because it predicts \(-167\) words.