2017 TWO(b)

Achievement
Question
The percentage of seeds germinating depends on the amount of water applied to the seedbed
that the seeds are sown in, and may be modelled by the function:

P(w)=96ln⁡(w+1.25)−16w−12P(w) = 96\ln(w + 1.25) - 16w - 12

where PP is the percentage of seeds that germinate and
ww is the daily amount of water applied (litres per square metre of seedbed), with 0≤w≤150 \le w \le 15.

Find the amount of water that should be applied daily to maximise the percentage of seeds
germinating.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*
Official Answer
P(w)=96ln⁡(w+1.25)−16w−12P(w)=96\ln(w+1.25)-16w-12
dPdw=96w+1.25−16\dfrac{dP}{dw}=\dfrac{96}{w+1.25}-16

Maximum when dPdw=0\dfrac{dP}{dw}=0

96w+1.25−16=0\dfrac{96}{w+1.25}-16=0
96=16(w+1.25)96=16(w+1.25)
76=16w76=16w
w=4.75w=4.75
Grading Criteria

Achievement (u)

  • Correct solution with correct derivative.

Merit (r)

-

Excellence T1

-

Excellence T2

-
Video Explanation
NCEA Level 3 Calculus Differentiation 2017 NZQA Exam - Worked Answers by infinityplusone(starts at 26:17)
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Exam Paper (2017)Assessment Schedule (2017)
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